Matrix #1:
[[7, 3, 0, 2, 8],
[5, 1, 9, 4, 0],
[2, 8, 3, 7, 1],
[6, 4, 0, 9, 5],
[1, 7, 2, 3, 8]]

Matrix #2:
[[0, 5, 8, 1, 7],
[3, 2, 9, 4, 6],
[7, 0, 3, 8, 1],
[5, 9, 2, 0, 4],
[8, 1, 7, 3, 2]]

Multiply these matrices according to mathematical rules and show the result.

<think>
Okay, so I need to multiply these two matrices. Hmm, matrix multiplication can be a bit tricky if you don't remember the steps. Let me start by recalling how matrix multiplication works.

First off, the number of columns in the first matrix has to match the number of rows in the second matrix for them to be compatible. Here both matrices are 5x5, so that's good—each will have 5 rows and 5 columns. The result should also be a 5x5 matrix.

The key part is remembering how each element in the resulting matrix is calculated. Each entry (i,j) in the product matrix is the sum of the products of elements from row i of the first matrix and column j of the second matrix. So, for example, the element at position (1,1) is computed by taking the first row of Matrix #1 and multiplying each element with the corresponding element in the first column of Matrix #2, then adding all those up.

Let me write down the steps more concretely. Let's denote Matrix #1 as A and Matrix #2 as B. The resulting matrix C will be such that:

C[i][j] = sum_{k=1 to 5} (A[i][k] * B[k][j])

Alright, let's start calculating each element step by step.

First row of Matrix #1 with columns of Matrix #2.

Starting with the first entry, C[0][0] (since matrices are often zero-indexed in my mind). Wait, actually, maybe I should think in 1-based indices since that's common for math problems. Let me clarify:

Assuming rows and columns start at 1 here, so element (1,1) is the first row first column.

Wait, actually, to avoid confusion, let's stick with zero-indexing because programming uses that, but maybe it doesn't matter here. Hmm, perhaps better just use 1-based as per standard matrix notation in math problems.

Let me see:

Matrix #1 (A):

Row 1: [7, 3, 0, 2, 8]

Column 1 of Matrix #2 (B):

[0,

3,

7,

5,

8]

So to get element C(1,1), multiply each corresponding elements and sum:

7*0 + 3*3 + 0*7 + 2*5 + 8*8.

Let me compute that step by step:

First term: 7*0 = 0

Second term: 3*3=9

Third term: 0*7=0 (this one doesn't contribute)

Fourth term: 2*5=10

Fifth term:8*8=64

Adding them up: 0 +9+0+10+64 = 83. So the first element is 83?

Wait, that seems okay.

Moving to C(1,2): which is row 1 of A and column 2 of B.

Column 2 in Matrix #2:

5,

2,

0,

9,

1

So compute 7*5 +3*2 +0*0 +2*9 +8*1.

Calculating each term:

7*5=35; 3*2=6; 0*0=0; 2*9=18; 8*1=8. Adding them up: 35+6=41, plus 18 is 59, +8=67.

So C(1,2)=67

C(1,3): row 1 of A * column3 of B.

Column3 of Matrix #2:

8,

9,

3,

2,

7.

Compute: 7*8 +3*9 +0*3 +2*2 +8*7.

Breaking down:

7*8=56; 3*9=27; 0*3=0; 2*2=4; 8*7=56. Total: 56+27 is 83, plus 4 is 87, plus 56 gives 143? Wait that's big. Let me check again:

Wait, 56 +27 =83? Yes.

Then 83 +0 (the third term) stays, then add 4: total 87, and 8*7 is 56. So 87+56=143. Hmm okay, maybe that's right.

C(1,4): Row 1 of A * column4 of B:

Column4 in Matrix #2:

1,

4,

8,

0,

3

Calculating:7*1 +3*4 +0*8 +2*0 +8*3

Compute each term:

7*1=7; 3*4=12; 0*8=0; 2*0=0; 8*3=24. Summing these gives:7+12=19, plus 0 and then 24 total is 43? Wait 19 + 24 = 43? Hmm yes.

Wait 7+12 is 19, adding the rest (which are zeros except for last term) so 19 +24 =43. Correct.

C(1,5): Row1 of A * column5 of B:

Column5 in Matrix #2: [7,6,1,4,2]

Wait let me confirm the columns again:

Matrix#2's first row is [0,5,8,1,7], so indeed column 5 entries are the fifth elements from each row.

So Column5 of Matrix B is:

First element (row1) is 7,

Second row (first element was 3?) Wait no. Wait second row starts with 3? Let me check again:

Matrix #2 as given by user:

Row1: [0, 5,8,1,7]

Row2: [3,2,9,4,6]

Row3:[7,0,3,8,1]

Row4: [5,9,2,0,4]

Row5:[8,1,7,3,2]

Therefore Column5 entries are:

First row's fifth element is 7,

Second row's fifth is 6,

Third is1,

Fourth is4,

Fifth is2.

Thus column5 of B: [7,6,1,4,2]

So computing C(1,5):

Row1 A elements multiplied by these terms.

7*7 (from first entry) +3*6 (second row's fifth element) +0*1 (third row's fifth?) Wait hold on. Let me confirm:

Wait actually, the third term is from the 3rd column of A? No, no. Let me rephrase: when taking Row i of A and Column j of B, so for each position k in the columns/rows.

So C(1,5):

Each element's product is (A[1][k] * B[k][5]) summed over k=1 to 5.

Therefore:

First term (k=1): A[1][1]*B[1][5] =7*7,

Second (k=2): A[1][2]*B[2][5]=3*6,

Third: A[1][3]*B[3][5]=0*1,

Fourth:A[1][4]* B[4][5]=2 *4,

Fifth: A[1][5] *B[5][5]=8*2.

Calculating each:

7*7=49; 3*6=18; 0*1=0; 2*4=8; 8*2=16. Adding up: 49+18=67, plus 8 gives 75, +16 is 91.

Thus C(1,5)=91.

Alright so first row of the result matrix is [83, 67, 143, 43, 91]. Hmm that's a lot. Let me verify some entries again just to be safe.

Wait for example C(1,3) was 143? Let's see:

First term: A[1][1]*B[1][3] =7*8,

Second term: A[1][2]*B[2][3]=3*9,

Third:0 * B[3][3] (which is 3, so 0*3),

Fourth: A[1][4]*B[4][3] is 2*2,

Fifth:8*7 (since column3 of row5 in B is 7? Wait wait:

Wait columns in matrix#2's third column.

First entry in column3: 8,

second row has 9 in column3,

third row has 3 in column3,

fourth row has 2,

fifth row has 7?

Yes, because looking at the third element of each row in Matrix #2:

Row1 third element is 8,

Row2's third element is 9,

Row3: third is3,

Row4: third is2,

Row5: third is7.

So for column3:

[8;9;3;2;7]

Therefore, yes, fifth term would be 8*7 (since A[1][5] =8, times B[5][3]=7).

Thus the terms are correct. So 7*8=56,

+3*9=27 → total so far:83,

0*3 adds nothing,

2*2=4 gives 87,

plus 8*7=56, totaling 143. Okay that's right.

Alright moving to the second row of matrix A and compute its dot product with columns of B for each element in C's second row.

So let me start with C(2,1):

Second row of A is [5,1,9,4,0]

Multiply by first column of B (which was the leftmost column entries as before).

First column of B:

Row1:0,

row2:3,

row3:7,

row4:5,

row5:8.

So:

5*0 +1*3 +9*7 +4*5 +0*8.

Compute each term:

5×0=0; 1×3=3; 9×7=63; 4×5=20; 0×8=0. Adding them:0+3=3, +63 is 66, +20 gives 86, plus nothing else (the last term). So total is 86? Wait:

Wait 3 +63 =66,

66 +20 is 86 yes.

Therefore C(2,1)=86?

Hold on let me check again. Hmm.

Yes:

5*0=0,

then first column's second element (row2) is 3 → 1×3,

third row of B's first column is7 →9×7,

fourth element in the column is5 →4×5,

fifth element is8 ×0, which adds nothing. So yeah.

So C(2,1)=86.

Next, C(2,2):

Second row of A [5,1,9,4,0] * second column of B (which is the middle column entries in the original problem's matrix 2 columns).

Wait let me confirm:

The second column of B is:

Row1:5,

row2:2,

row3:0,

row4:9,

row5:1.

Thus, compute each term:

5*5 (since first element A[2][1] =5 times B's row1's col2=5),

then 1×2 (A[2][2]*B[2][2]),

+9*0 (from third entry of second column in B),

+4×9,

+0×1.

Calculations:

First term: 5*5 =25,

Second:1×2=2 → total so far 27,

Third term adds nothing (since it's 9×0),

Fourth term:4×9=36 → total becomes 27 +36=63,

Fifth term:0×1=0. Total is 63? Wait adding up all terms:

25+2=27; 27 +0 remains, then add 36 (total 63), plus zero gives 63.

Thus C(2,2)=63.

C(2,3): second row of A times column3 of B. Column3 as before was [8,9,3,2,7].

Wait let me write the terms:

5×8 (A[2][1]=5 * B[1][3}=8),

+1×9 (second element: 1 ×B's second row third col is9),

+9×3 (third column entry in B for row3 is3? Wait yes, third row's third column was 3,

then fourth term:4×2 (fourth row third column of B is 2),

and fifth term:0*7.

Calculating:

5×8=40,

1×9=9 → total so far 49,

+9×3=27, so now 49 +27 =76,

plus 4×2=8 → total 76+8=84,

then the last term is zero.

Total: 40+9+27+8=84? Wait let's recompute:

Wait 5*8 is 40,

1*9=9 (total now 49),

plus 9×3 =27 → total 76,

then +4×2=8 → 76+8=84,

Yes. Then the last term is 0, so total 84.

So C(2,3)=84?

Wait hang on: 4*2? Let me confirm:

The fourth entry in column3 of B (fourth row third element) is indeed 2 (from Matrix #2's fourth row [5,9,2,0,4]), so yes. So that term is okay.

Proceeding to C(2,4): second row A times column4 of B.

Column4 in B:

First entry:1,

row2:4,

row3:8,

row4:0,

row5:3.

Thus terms are:

5×1 (A's first element 5 × B[1][4]=1),

plus 1×4 (second elements, A[2] is row second? Wait no, in the second row of A, so entries are [5,1,9,4,0], so column indices k from 1 to 5.

Wait let me structure it properly:

Each term is A's row2 elements multiplied by B's column4 (which has rows as above):

A[2][1]*B[1][4] =5*1,

A[2][2]*B[2][4]=1×4,

A[2][3]*B[3][4}=9×8,

A[2][4}*B[4][4}=4×0,

A[2][5]*B[5][4}=0×3.

So calculating:

First term:5*1=5,

Second term 1×4=4 → total so far 9,

Third term:9×8=72 → total now 81,

Fourth term:4×0=0,

Fifth term:0×3=0.

Total sum is 5+4 +72 = 81.

Thus C(2,4)=81?

Wait that seems okay:

Adding the terms: 5+4 is 9 plus 72 gives 81. Yes.

Now moving to C(2,5): second row of A [5,1,9,4,0] multiplied by column5 of B (which was [7,6,1,4,2]).

Calculations:

Terms are 5×7 (first element),

+1×6,

+9×1,

+4×4,

+0×2.

Compute each term:

5*7=35;

1×6=6 → total so far:41,

9×1=9 → total now 50,

4×4=16 → 50 +16 =66,

0×2 adds nothing. Total is 66?

Wait let's add again:

First term 35,

plus 6 (total 41),

+9 gets to 50,

then +16 gives 66, yes.

So C(2,5)=66.

Alright second row of the resulting matrix so far: [86,63,84,81,66]

Moving on to third row of A (row3) which is [2,8,3,7,1]. Let's compute its dot products with each column in B.

Starting with C(3,1): row3 of A * first column of B:

First column entries are as before:0, 3, 7,5,8.

So terms:

2×0 (from A[3][1] and B[1][1]),

+8×3,

+3×7,

+7×5,

+1×8.

Calculations:

2*0=0,

8×3=24,

3×7=21,

7×5=35,

1×8=8.

Adding them up: 0 +24 is24, plus 21 →45, +35 =80, +8 gives 88.

Thus C(3,1)=88?

Wait check again:

Let me add step by step:

First term 0,

then +24=24,

+21 (total 45),

+35 → total 80,

plus the last term 8 =88. Yep.

C(3,2): third row of A [2,8,3,7,1] multiplied by column2 of B:

Column2: entries are 5 (row1),2(row2),0 (row3),9 (row4),1 (row5).

So terms:

2×5,

+8×2,

+3×0,

+7×9,

+1×1.

Calculating each term:

2*5=10,

8×2=16 → total 26 so far,

3×0=0,

then +7×9=63 (total becomes 26+0+63 =89),

plus 1×1=1. Total: 10+16+0+63+1=90?

Wait step by step:

First terms:

10 (from first term) +16 is 26,

then +0 remains 26,

adding 7×9 which is 63 → total 89,

plus the last term 1 gives 90. So C(3,2)=90.

Proceeding to C(3,3):

Third row of A and column3 of B:

Column3 is [8,9,3,2,7].

Thus terms are:

2×8 (first element),

+8×9,

+3×3,

+7×2,

+1×7.

Calculating each:

First term 2×8=16,

second:8*9=72 → total so far 88,

third term:3×3=9 → adding to 97,

fourth term:7×2=14 → total 111,

last term:1×7=7 gives final total of 118?

Wait:

Let me add again:

16 (first) +72 = 88,

+9 →97,

+14 makes 111,

+7 →118. Correct.

Thus C(3,3)=118?

Okay.

C(3,4): third row of A multiplied by column4 of B:

Column4's entries are [1 (row1),4(row2),8(row3),0 (row4), 3 (row5)]? Wait let me check matrix #2 again for the fourth column:

Looking at Matrix #2:

Row1:0,5,8,1,7 → so fourth element is1,

Row2:3,2,9,4,6 → fourth is4,

Row3:7,0,3,8,1→ 8 in column4? Wait no. Wait original matrix#2's row3:

[7 (col1), 0(col2?), wait let me confirm:

Wait Matrix #2 as given was:

[[0,5,8,1,7], → first row,

[3,2,9,4,6], second row,

[7,0,3,8,1], third row: so yes the fourth column entry here is 8.

Row3's fourth element is 8,

then Row4 of matrix#2 is [5,9,2,0,4] → fourth column is 0,

and fifth row: [8,1,7,3,2]. So fourth column element for row5 is 3.

Thus column4 is:

[1 (row1),4(row2),8(row3),0(row4),3(row5)].

Therefore terms for third row A and column4 are:

A's elements: [2,8,3,7,1] multiplied by B's entries in column4:

First term 2×1,

then 8×4,

+3×8 (third element of column4 is8),

+7×0,

+1×3.

Compute each term:

2×1=2,

8×4=32 → total so far:34,

next term:3×8=24 → now 34 +24 =58,

then 7×0 adds nothing,

and 1×3=3. Total is 58+3=61.

Wait summing all:

First term 2,

+32 (total 34),

+24 (now 58),

plus 3 gives total of 61. So C(3,4)=61?

Yes.

Proceeding to C(3,5):

Third row of A and column5 of B (the fifth column entries are [7,6,1,4,2] as per matrix#2's rows:

Row1:...,7,

row2:...,6,

row3:...,1,

row4:...,4,

row5:...,2.

So terms:

A elements multiplied by these:

First term 2×7,

+8×6,

+3×1,

+7×4,

+1×2.

Calculating:

2×7=14,

8×6=48 → total so far 62,

next:3×1=3 → 65,

then +7×4=28 → 93,

plus 1×2 adds 2, totaling 95?

Wait let's verify:

First term:14,

+48 =62,

+3 gives 65,

+28 (from 7×4)=93,

+2 → total 95.

Thus C(3,5)=95.

So third row of the resulting matrix is [88,90,118,61,95].

Moving to fourth row of A: [6,4,0,9,5]. Let's compute its products with each column in B.

Starting with C(4,1): fourth row of A times first column:

First column entries are 0 (row1),3(row2),7(row3),5(row4),8(row5).

Thus terms:

6×0,

+4×3,

+0×7,

+9×5,

+5×8.

Calculations:

First term: 6*0=0,

then +4×3=12 → total so far 12,

third term is zero (since A's third element here is zero),

next term:9×5=45 → adding gives 57,

last term:5×8=40. Total sum: 12+45=57 +40 =97.

So C(4,1)=97?

Check again:

First term:0,

Second term 4*3=12,

then 0,

next term 9×5 is 45 → 12+45=57,

plus last term 5×8=40, so total 57+40=97. Correct.

C(4,2): fourth row A times column2 of B:

Column2 entries are:5 (row1),2(row2),0(row3),9(row4),1(row5).

Terms:

6×5,

+4×2,

+0×0,

+9×9,

+5×1.

Calculating each term:

First term 6*5=30,

then +4×2=8 → total so far 38,

third term is zero (since A's third element here is 0),

then 9×9=81 → adding to get 38+81=119,

plus 5×1=5, totaling 119 +5 =124.

Wait step by step:

First term:30,

second: add 8 → 38,

third term zero,

then 81 added (total becomes 38+81=119),

last term adds 5 → total is124.

Therefore, C(4,2)=124.

C(4,3): fourth row A [6,4,0,9,5] multiplied by column3 of B (column3 entries are 8(row1),9(row2),3(row3),2(row4),7(row5)):

Terms:

6×8,

+4×9,

+0×3,

+9×2,

+5×7.

Compute each term:

First:6*8=48,

second:4×9=36 → total so far 84,

third term is zero,

then 9×2=18 → total becomes 84 +18 =102,

last term:5×7=35. So adding those two gives 102+35=137?

Wait:

48 (first) +36=84,

plus zero remains,

then +18=102,

+35 → total is 137.

Thus C(4,3)=137?

Yes.

C(4,4): fourth row A times column4 of B:

Column4 entries are [1 (row1),4(row2),8(row3),0 (row4),3(row5)].

Terms:

6×1,

+4×4,

+0×8,

+9×0,

+5×3.

Calculations:

First term 6×1=6,

second:4×4=16 → total so far 22,

third term is zero (since A's third element here is zero),

then 9×0 adds nothing,

last term:5×3=15.

Total sum: 6+16 +15 =37.

Wait let me verify:

First term:6,

plus second term 16 →22,

third term (0*8)=0,

fourth term (9×0)=0,

last term:5×3=15.

Total is 22+0+15=37? Wait yes, because the fourth column of B's row4 is 0 here. So correct.

Therefore C(4,4)=37?

Wait hang on, let me check again:

Wait matrix #2's rows for column4:

Row4 (fourth row in matrix#2) is [5,9,2,0,4], so the fourth element there is 0. Therefore yes, that term is zero.

So total:6+16 +15 =37? Wait, adding 6+16=22; 22+0 (third and fourth terms) plus 15 gives 22+15=37. Correct.

C(4,5): fourth row A multiplied by fifth column of B:

Fifth column entries are [7(row1),6(row2),1(row3),4(row4),2(row5)].

Terms:

6×7,

+4×6,

+0×1,

+9×4,

+5×2.

Calculations:

First term 6×7=42,

second term:4×6=24 → total so far 66,

third is zero (since third element of A row here is zero),

then +9×4=36 → total now 66+36 =102,

last term:5×2=10. Total sum 102+10=112.

Thus C(4,5)=112?

Wait let's check:

First term 42,

plus second (24) gives 66,

third term adds zero,

fourth term 36 → total up to that point:66+36=102,

last term 10. Total is indeed 112.

Okay, fourth row of the result is [97, 124, 137, 37, 112].

Proceeding to fifth and final row (row5 of matrix A):

Matrix #1's fifth row: [1,7,2,3,8].

Now compute each element in this row multiplied with corresponding column entries from Matrix#2.

Starting with C(5,1): fifth row times first column:

First column entries are 0 (row1),3(row2),7(row3),5(row4),8(row5).

Terms:

1×0,

+7×3,

+2×7,

+3×5,

+8×8.

Calculating each term:

First:0,

second term:7*3=21,

third:2×7=14 → total so far 21+14=35,

fourth term:3×5=15 → adds to make 50,

fifth term:8×8=64. Total sum is 35 +15 =50 +64=114.

Wait step by step:

First term (1*0)=0,

second:21,

third:14 → total after three terms is 0+21+14=35,

fourth term 15 gives 35+15=50,

plus last term 64 makes 114.

So C(5,1)=114?

Wait hang on:

Yes.

C(5,2): fifth row A times column2 of B (column entries [5(row1),2(row2),0(row3),9(row4),1(row5)]):

Terms:

1×5,

+7×2,

+2×0,

+3×9,

+8×1.

Compute each term:

First:1*5=5,

second:7×2=14 → total so far 19,

third term is zero (since 2×0),

fourth term:3×9=27 → adds to get 19+27=46,

last term:8×1=8. Total sum:46+8=54.

Thus C(5,2)=54?

Wait:

Yes, 5+14 +0 +27 +8 = 54.

C(5,3): fifth row A multiplied by column3 of B (column entries [8(row1),9(row2),3(row3),2(row4),7(row5)]):

Terms are:

1×8,

+7×9,

+2×3,

+3×2,

+8×7.

Calculations:

First term:8,

second:63,

third:2*3=6 → total so far 8+63+6=77,

fourth term:3×2=6 (total now 83),

last term:56. Total sum is 77 +6 =83 +56=139?

Wait:

First terms:

8 (from first),

plus second term 63 → total 71,

third term adds 6 to get 77,

fourth term adds 6 → 83,

last term:56. Total is 83+56=139.

Thus C(5,3)=139?

Yes.

C(5,4): fifth row A and column4 of B (column entries [1 (row1),4(row2),8(row3),0(row4),3(row5)]):

Terms are:

1×1,

+7×4,

+2×8,

+3×0,

+8×3.

Calculating each term:

First:1*1=1,

second:28 → total so far 29,

third:16 (from 2×8) adds to give 45,

fourth term is zero,

last term:24. Total sum is 45 +0 +24 =69?

Wait:

First terms:

1,

+28=29,

then third term 16 gives total 29+16=45,

plus fourth (zero),

and last term 24 → total 45 +24=69. Correct.

So C(5,4)=69?

Yes.

C(5,5): fifth row A multiplied by fifth column of B (which has entries [7(row1),6(row2),1(row3),4(row4),2(row5)]):

Terms are:

1×7,

+7×6,

+2×1,

+3×4,

+8×2.

Calculating each term:

First:7,

second:42 → total so far 49,

third term:2*1=2 (total now 51),

fourth term:12 adds to make 63,

last term:16. Total sum is 51 +12 +16 =?

Wait let's do it step by step:

First term:7,

second term adds 42 → total 49,

third term adds 2 → 51,

fourth term (3×4=12) gives 51+12=63,

last term is 8×2=16 → total becomes 63 +16 =79.

Wait wait, that seems off. Let me check again:

Wait first term:1×7=7,

then second term is7×6=42 → total so far 49,

third term: 2×1=2 (total now 51),

fourth term:3×4=12 (now 63),

last term:8×2=16. Total of all terms:7+42+2+12+16.

Adding step by step:

7 +42 =49,

49 +2=51,

51+12=63,

63+16=79. Yes, correct.

So C(5,5)=79.

Thus fifth row of the resulting matrix is [114, 54, 139, 69,79].

Putting all rows together, the result should be a 5x5 matrix:

Row1: Let me first check my previous computations again step by step to ensure I didn't make any errors. But proceeding with what I have.

Wait wait, hang on. Wait in computing the first row of the multiplication, which is Row1 of A multiplied by each column of B. However above I started with the fourth and fifth rows, but for thoroughness, I need to compute all elements step-by-step properly.

**Hold on! There's a problem here. I see now that when starting calculations for the result matrix, in my initial steps I considered the first row as Row1 of A multiplied by columns of B, however in the process above, after working out rows 4 and 5, but skipped rows1-3? Wait actually looking back:

Wait let me check again. The user's question is to multiply Matrix #1 (A) with Matrix #2 (B), following mathematical rules.

The steps I did were for Rows4 and Row5 of the resulting matrix, but in my initial part above, after setting up the problem, I wrote:

"Matrix #1: [[7, 3, 0, 2, 8], ..."

Then proceeded to compute entries starting with row4 and row5. However I missed rows1-3.

Wait that's a mistake! So actually in my initial calculation above, I computed only rows 4 and 5 of the result matrix. Rows1,2,3 were not calculated yet. That's an error.

Therefore, need to redo the calculations properly for all elements.

Okay, let me start over methodically:

To compute each element (i,j) in the resulting matrix C = A × B,

C[i][j] is the dot product of row i from Matrix A and column j from Matrix B.

Given that both matrices are 5x5. The result should also be a 5x5 matrix.

Let me proceed step by step for each element:

First, I'll write down both matrices again clearly.

Matrix #1 (A):

Row1: [7,3,0,2,8]

Row2: [5,1,9,4,0]

Row3: [2,8,3,7,1]

Row4: [6,4,0,9,5]

Row5: [1,7,2,3,8]

Matrix #2 (B):

Column1: [0, 5,7,5,8] (elements from each row's first column)

Wait actually to get the columns of B, need to transpose the rows:

Actually the structure is:

B's first row is [0,5,8,1,7], so its columns are as follows:

Column1: 0 (from first row), then second element in column1 is second row's first element: 3,

Third element of column1 is third row's first element:7,

Fourth row's first element:5,

Fifth row's first element:8.

So Column1 of B is [0,3,7,5,8]

Column2 of B:

First element from first row second column:5,

Then 2 (second row's second),

Third row's second:0,

Fourth row's second:9,

Fifth row's second:1. So column2 is [5,2,0,9,1].

Column3 of B:

[8 (from first),9 (second),3(third),2(fourth),7(fifth)]. Wait no:

Wait first row third element of B is 8,

Second row third:9,

Third row third:3,

Fourth row third:2,

Fifth row third:7. So column3 is [8,9,3,2,7].

Column4 of B:

First row fourth element:1,

Second row fourth:4,

Third row fourth:8,

Fourth row fourth:0,

Fifth row fourth:3 → column4 is [1,4,8,0,3]

Column5 of B:

First row fifth element:7,

Second row fifth:6,

Third row fifth:1,

Fourth row fifth:4,

Fifth row fifth:2 → so column5 is [7,6,1,4,2].

Now proceeding.

Starting with computing all elements step by step for resulting matrix C.

Let me structure it as:

C will have entries from (Row1Col1) to (Row5Col5).

First, compute Row 1 of the result:

For row i=1 (first row of A):

Compute each column j=1 to 5 (columns of B)

So,

C[1][1] = (A's Row1 • B's Column1):

7*0 +3*3 +0*7 +2*5 +8*8 → compute term by term:

7×0=0

3×3=9

0×7=0

2×5=10

8×8=64

Sum: 0+9+0+10+64 = 83.

Wait, let's calculate again step by step to prevent error:

Row1 of A is [7,3,0,2,8]

Column1 of B is [0 (from first row), 3 (second),7 (third),5(fourth),8(fifth)]

Dot product:

(7 * 0) + (3*3)+(0*7)+(2*5)+(8*8)

= 0 +9+0+10+64 = total of 83. So C[1][1]=83.

C[1][2] = Row1 of A • Column2 of B:

Column2 is [5,2,0,9,1]

So,

7×5 +3×2 +0×0 +2×9 +8×1

Calculate each term:

7*5=35

3×2=6 → 35+6=41

0×0=0

2×9=18 → total now 41+0+18=59

8×1=8 → total sum is 59+8=67.

Therefore C[1][2] =67?

Wait:

Compute each term step by step:

35 (from first) +6 =41,

plus 0 gives still 41,

then plus 2×9=18: 41+18=59,

plus last term (8×1)=8 → total 59+8=67. Yes.

C[1][3] = Row1 • Column3 of B:

Column3 is [8,9,3,2,7]

Thus,

7×8 +3×9 +0×3 +2×2 +8×7

Compute term by term:

7*8=56,

3×9=27 → total so far 56+27=83,

then 0×3=0 → remains 83,

next 2×2=4 (total now 83+4=87),

plus last term:8×7=56 → 87 +56 =143.

So C[1][3]=143?

Wait:

Yes, adding all together:

56 +27 is 83,

then add 0 (still 83),

+4 gives 87,

plus 56 gives 143. Correct.

C[1][4] = Row1 • Column4 of B:

Column4: [1,4,8,0,3]

So,

7×1 +3×4 +0×8 +2×0 +8×3

Compute terms:

7*1=7,

3×4=12 → 7+12=19,

0×8=0 (remains 19),

2×0=0 (still 19),

8×3=24. Total is 19 +24=43.

Thus C[1][4]=43?

Yes:

7+12=19, plus others: 0,0 → total so far still 19,

plus 24 (from last term) gives 43.

C[1][5] = Row1 • Column5 of B:

Column5 is [7,6,1,4,2]

Calculations:

7×7 +3×6 +0×1 +2×4 +8×2

Compute terms:

7×7=49,

3×6=18 → 49+18=67,

0×1=0 (still 67),

2×4=8 → total now 75,

plus last term:8×2=16. Total is 75 +16 =91.

Thus C[1][5]=91?

Yes:

49+18=67,

plus 0 stays,

then add 8 (now 75),

plus 16 gives total of 91. Correct.

So first row of Result Matrix is [83, 67, 143, 43, 91].

Now moving on to Row2 of the resulting matrix (second row of A):

Row2 of A: [5,1,9,4,0]

Compute for each column j=1 to5.

C[2][1] = Row2 • Column1:

Column1 elements again are [0,3,7,5,8].

Calculations:

5×0 (first term) + 1×3 (second element of B's column) +

9×7 (third),

4×5 (fourth),

0×8 (fifth).

So,

5*0=0,

1×3=3 → total so far 0+3=3,

then +9×7=63 → 66,

plus 4×5=20 → total now 66+20=86,

plus last term (0×8)=0. Total is 86.

Thus C[2][1]=86.

C[2][2] = Row2 • Column2:

Column2 elements are [5,2,0,9,1].

Calculations:

5×5 +1×2 +9×0 +4×9 +0×1

Compute each term:

5×5=25,

1×2=2 → total 27 so far,

+9*0=0 → remains 27,

then 4×9=36 → total now 63,

+0×1=0. Total is 25 +2 =27, plus 36 gives 63.

Wait step by step:

First term:5×5=25,

Second term:1×2=2 → sum to 27,

Third term (9×0)=0 → total still 27,

Fourth term 4×9=36 → adds up to 27+36=63,

Fifth term:0×1=0. Total is 63.

Thus C[2][2] =63.

C[2][3]= Row2 • Column3:

Column3 elements [8,9,3,2,7].

Calculations:

5×8 +1×9 +9×3 +4×2 +0×7

Compute term by term:

5×8=40,

1×9=9 → total 49,

+9×3=27 → now 76,

+4×2=8 (total 84),

+0×7=0. Total: 40+9 is 49, plus 27 gives 76, +8=84.

Thus C[2][3]=84?

Wait:

Yes 40+9+27+8 = 84 (since the last term is zero).

Wait let me verify again:

5×8=40,

1×9=9 → total so far 49,

then 9×3=27, which sums to 49 +27=76,

plus 4×2=8 gives 84,

then last term is zero. So yes, 84.

C[2][4] = Row2 • Column4:

Column4 elements are [1,4,8,0,3].

Thus,

5×1 +1×4 +9×8 +4×0 +0×3

Compute each term:

5×1=5,

1×4=4 → total 9 so far,

+9×8=72 (now total 81),

then 4×0=0 remains at 81,

and last term 0×3=0. So total is 5+4+72=81.

C[2][4]=81.

C[2][5] = Row2 • Column5:

Column5 elements [7,6,1,4,2].

Thus,

5×7 +1×6 +9×1 +4×4 +0×2

Compute each term:

5×7=35,

1×6=6 → sum 41,

+9×1=9 → now 50,

+4×4=16 (total 66),

+0×2=0.

Total: 35+6=41, +9→50, +16→66.

Thus C[2][5] =66?

Yes:

35+6 is 41,

plus 9 →50,

plus 16 (from 4×4) gives total of 66.

So second row entries: [86,63,84,81,66].

Now Row3 of Result Matrix (Row3 from A):

Row3 elements are [2,8,3,7,1]

Compute for each column j=1 to5.

C[3][1] = Row3 • Column1:

Column1: [0,3,7,5,8]

Calculation:

2×0 +8×3 +3×7 +7×5 +1×8

Compute each term:

2*0=0,

8×3=24 → total so far 24,

+3×7=21 (total now 45),

then 7×5=35 → adds to 45→80,

and finally 1×8=8. Total sum: 24+21=45, +35=80, +8=88.

Therefore C[3][1]=88?

Wait:

Let me check again step by step:

First term (Row3 first element * Column1 first) =2×0=0,

Second term: 8 (second elem of row3) * column1 second element(3)=24,

Third term: 3*7 (third elements: third in row, third in column?) Wait yes.

Wait the row is [2,8,3,7,1] so first element is 2, then next is 8 etc. The Column1 has entries from each row's first element:

Column1 for B is [0 (row1), 3(row2),7(row3),5(row4),8(row5)].

Thus,

Row3 of A is interacting with all elements in column1:

First term: Row3, element1 (which is 2) × Column1's first element (B_row1_col1=0),

Second term: Row3 element2 (8) × B_row2_col1 (3),

Third term: Row3 element3 (3) × B_row3_col1(7),

Fourth term: Row3 element4 (7) × B_row4_col1(5),

Fifth term: Row3 element5 (1) × B_row5_col1(8).

Therefore:

2×0 = 0,

8×3=24,

3×7=21,

7×5=35,

1×8=8.

Adding all together: 0 +24+21+35+8 → total is 24+21=45, 45+35=80, 80+8=88. Correct.

C[3][2] = Row3 • Column2 of B:

Column2 elements are [5 (row1),2(row2),0(row3),9(row4),1(row5)].

Calculations:

Row3 elements:2,8,3,7,1 multiplied by column2's elements respectively:

2×5 +8×2 +3×0 +7×9 +1×1

Compute each term:

2×5=10,

8×2=16 → 26 so far,

+3×0=0 (still 26),

then 7×9=63 → total now 89,

plus last term:1×1=1. Total is 89 +1 =90.

Therefore C[3][2]=90?

Check:

Yes, step by step:

First term 2×5=10,

Second: 8*2=16 → 26,

Third: 0 → remains,

Fourth:7*9=63 gives total up to 89,

Last term +1. Total is 90.

C[3][3]= Row3 • Column3:

Column3 elements [8,9,3,2,7].

Thus,

Row3 elements multiplied by column3's elements:

2×8 (from row1) +

8×9 (row2 col3),

3×3 (third element both),

7×2 (fourth),

1×7 (fifth).

Compute term by term:

2×8=16,

8×9=72 → 16+72=88,

3×3=9 → total now 97,

then +7×2=14 → 111,

plus last term:1×7=7. Total is 16+72=88, +9=97, +14=111, +7=118.

Wait wait let me compute again:

First term (Row3's first element × B_row1_col3)= 2×8,

Second: Row3 second × B's row2 col3:8×9,

Third: third elements (both are in the same position here? Wait no, each element is from different rows.

Wait Column3 of B has:

First element (row1) is 8,

second (row2) is 9,

third (row3) is3,

fourth (row4) is2,

fifth (row5) is7.

So the terms are indeed as above:

2×8 =16,

8×9=72 → total so far 88,

3×3 (since third element of row3 multiplied by B's column3 row3)=3×3=9 → adds to 88→97,

then fourth term: 7 (fourth elem in Row3) × column3 fourth entry (which is2 from row4): 7×2=14, total now 97+14=111,

last term:1×7=7 → total 118.

Yes so C[3][3] =118?

Wait but let me recalculate once more:

First term 2*8=16,

Second term 8*9=72 (total 88),

Third term 3*3=9 (now 97),

Fourth term 7×2=14 → total 111,

Fifth term 1×7=7. Total is 16+72=88, plus 9 makes 97; plus 14 gives 111, then +7 → 118. Yes.

C[3][4] = Row3 • Column4:

Column4 elements are [1 (row1),4(row2),8(row3),0(row4),3(row5)].

Calculations:

Row3's elements × column4 entries:

2×1 +8×4 +3×8 +7×0 +1×3

Compute each term:

2×1=2,

8×4=32 → total 34 so far,

+3×8=24 (total now 58),

then +7×0=0 remains at 58,

and finally 1×3=3. Total sum is 58 +3 =61.

So C[3][4]=61?

Verify:

Yes: first term 2×1=2,

second:8×4=32 → total 34,

third:3×8=24 (now 58),

fourth:7×0=0,

fifth:1×3=3. Total is indeed 61.

C[3][5] = Row3 • Column5:

Column5 elements [7,6,1,4,2].

Calculations:

Row3 elements × column5 entries:

2×7 (row1) +

8×6 (row2),

3×1,

7×4,

1×2.

Compute each term:

First: 2×7=14,

Second:8×6=48 → total so far 62,

Third:3×1=3 → total 65,

Fourth:7×4=28 (now 93),

Fifth:1×2=2. Total sum is 14+48=62 +3→65, +28→93, plus 2 gives 95.

Wait let's check again:

First term:2*7=14,

Second term:8×6=48 → total 14+48=62,

Third term (third element of row3 which is3 × column5 third entry, which is B_row3_col5: the original Matrix #2's third row fifth element is 1. So yes 3×1=3. Total now 65.

Fourth term: seventh element (row3 fourth element is7) multiplied by column5's fourth element (B's row4 col5 is4). Thus, 7*4=28 → total 65+28=93,

Fifth term:1×2=2 → total 95.

Thus C[3][5]=95.

Therefore third row of result matrix is [88,90,118,61,95].

Moving to Row4 of Result Matrix (Row4 from A):

Matrix#1's Row4: [6,4,0,9,5]

Compute each column from 1 to5:

C[4][1] = Row4 • Column1 of B.

Column1 is [0,3,7,5,8].

Calculations:

6×0 (row1) +

4×3 (row2),

0×7 (row3),

9×5 (row4),

5×8 (row5).

Compute each term:

First: 6*0=0,

Second:4×3=12 → total so far 12,

Third:0×7=0,

Fourth:9×5=45 → total now 57,

Fifth:5×8=40 → adds to 57+40=97.

So C[4][1]=97?

Wait:

Let's verify each term:

Row4 of A has elements [6,4,0,9,5]. Each element is multiplied by the corresponding row in column1 (which is from B matrix rows):

First entry: Row4,A first elem × B_row1_col1 (0): 6×0=0,

Second term:4×3 (from B_row2_col1)=12,

Third term:0×7 (B_row3_col1)=0,

Fourth term:9×5 (B_row4_col1)=45,

Fifth term:5×8 (B_row5_col1)=40.

Summing all terms:

0+12=12, +0 →12, +45→57, +40→97. Correct.

C[4][2] = Row4 • Column2 of B:

Column2 elements are [5(row1), 2(row2), 0(row3),9(row4),1(row5)].

Calculations:

6×5 (row1) +

4×2 (row2),

0×0 (row3's column2? Wait,

Wait, Column2 of B has first element 5 (from row1), second 2 (row2), third is from row3 which in Matrix#2 column2 is 0 (given in Matrix #2 above as row3: [7,0,...], so yes third entry 0).

Thus:

6×5,

4×2,

0×0,

9×9 (wait wait no. Wait the fourth term is Row4,A's element which is9 multiplied by B_row4's column2 entry.

Column2 of B for row4 is9? Let me confirm Matrix#2:

Matrix #2 second matrix as given is:

Row1: [0,5,8,1,7] → columns 1-5,

Wait let me recheck the matrices to avoid errors.

Original problem says Matrix #2:

[[0,5,8,1,7],

[3,2,9,4,6],

[7,0,3,8,1],

[5,9,2,0,4],  --> fourth row of matrix#2: columns are [5,9,2,0,4]. So column2 entry is9 for row4.

Row5 of Matrix #2:

[8,1,7,3,2].

So yes, the entries in Column2 (second column) are:

First row:5,

Second:2,

Third:0,

Fourth:9,

Fifth:1.

Thus returning to C[4][2] calculation:

Row4,A elements × column2 of B:

6×5 (row1's col2 is5),

+4×2 (B_row2_col2=2),

+0×0 (since third element in A row4 is0, and B_row3_col2 is0),

+9×9 (since 4th term from B_row4_col2=9),

+5×1 (fifth entry: A's fifth element5 × B_row5_col2 which is1).

Compute each:

6×5=30,

4×2=8 → total so far 38,

0×0=0 (still 38),

9×9=81 → 38+81=119,

then +5×1=5. Total: 30+8=38, +81→119 +5=124.

Thus C[4][2]=124?

Let me recheck:

First term (6×5)=30,

Second (4×2)=8 → total 38,

Third term:0×0=0,

Fourth (9×9)=81. So adding to previous total 119,

Fifth term (5×1)=5 added makes 124. Correct.

C[4][3] = Row4 • Column3:

Column3 entries of B are [8(row1),9(row2),3(row3),2(row4),7(row5)].

Calculations:

Row4,A elements:

6× first element (B row1 col3:8),

4× B_row2_col3=9,

0 × B_row3_col3=3,

9× B_row4_col3=2,

5× B_row5_col3=7.

Thus terms:

6×8,

4×9,

0×3,

9×2,

5×7.

Calculations step by step:

First term:6*8=48,

Second term 4×9=36 → total 48+36=84,

Third term is zero,

Fourth term:9×2=18 (total now 84+18=102),

Fifth term5×7=35 → total 102 +35 =137.

Therefore C[4][3]=137?

Check:

Yes, 6*8=48,

then 4×9 is 36 added (total 84),

next terms: 0+18 from 9×2 and 5×7=35. So yes total 137.

C[4][4] = Row4 • Column4 of B:

Column4 in Matrix#2 has entries [1(row1),4(row2),8(row3),0(row4),3(row5)].

Calculations:

Row4,A elements × column4:

6×1 (row1),

4×4,

0×8,

9×0,

5×3.

Compute each term:

First:6×1=6,

Second:4×4=16 → total so far 22,

Third:0×8=0,

Fourth:9×0=0,

Fifth:5×3=15. Total is 22+15=37.

Thus C[4][4]=37?

Yes:

First term6*1=6,

Second term 4×4=16 → total 22,

then +0, +0 and finally 5×3=15: total 37. Correct.

C[4][5] = Row4 • Column5 of B:

Column5 entries are [7(row1),6(row2),1(row3),4(row4),2(row5)].

Calculations:

Row4,A elements × column5:

6×7 (row1),

4×6,

0×1,

9×4,

5×2.

Compute each term:

First:6*7=42,

Second 4×6=24 → total so far 66,

Third:0×1=0,

Fourth:9×4=36 (total now 66+36=102),

Fifth:5×2=10. Total sum is 102 +10=112.

Therefore C[4][5] =112?

Checking:

Yes, 42 +24=66,

plus (9×4)=36 → total 102, plus 10 gives 112. Correct.

Thus row4 of result is [97,124,137,37,112].

Now moving to Row5 of Result Matrix, which corresponds to matrix#1's fifth row: [1,7,2,3,8].

Compute each column 1 through 5:

C[5][1] = Row5 • Column1 of B.

Column1 entries are [0(row1),3(row2),7(row3),5(row4),8(row5)].

Calculations:

Row5,A elements × column1:

1×0 (row1),

7×3,

2×7 (row3's column1 is7? Wait, yes for B matrix row3 column1 it's 7 as given in Matrix #2: third row starts with 7. So:

Wait let me confirm:

Matrix#2 first column entries are:

Row1:0,

Row2:3,

Row3:7,

Row4:5,

Row5:8.

Thus, yes:

Calculations for C[5][1]:

First term:1×0=0,

Second term 7×3=21,

Third term2×7 (since B_row3_col1 is7) =14,

Fourth term3×5 (from A's fourth element × row4 of column1)=3×5=15,

Fifth term8×8 (A fifth element × B_row5_col1 which is8).

Thus terms:

0 +21 +14 +15 +64.

Wait let me compute step by step:

First: 1*0 =0,

Second:7×3=21 → total so far 21,

Third:2×7=14 (total now 35),

Fourth term: third element in Row5,A is2? Wait, no. Wait the row is [1,7,2,3,8]. So the fourth element is3 and fifth is8.

Wait fourth element of matrix#1's fifth row is3:

Thus,

First term (Row5,A first elem × B_row1_col1) = 1×0=0,

Second: Row5,A second elem(7) × B_row2_col1(3): 7×3=21,

Third: third element of Row5 is2 → times B_row3_col1 which is7: 2×7=14,

Fourth element (3) times B_row4_col1 (5):3×5=15,

Fifth element (8) × B_row5_col1(8)=64.

Summing all terms:

0+21 +14 =35; 35 +15 is 50, then +64 → total of 114? Wait:

Wait let me do again:

Start at 0,

+21=21,

+14 →35,

+15→50,

+64 (which comes from last term:8×8)=50+64=114.

Thus C[5][1] =114?

Wait yes. Let me confirm again:

Row5 of A: [1,7,2,3,8]

Column1 of B:

Row1:0,

Row2:3,

Row3:7,

Row4:5,

Row5:8.

Hence the products are indeed 1×0=0; 7*3=21;2×7=14;3×5=15;8×8=64. Sum: 0+21+14+15+64 = (21+14)=35, +15=50, +64=114. Correct.

C[5][2] = Row5 • Column2 of B:

Column2 in matrix #2 is [5(row1), 2(row2),0,row3;9(row4);1(row5)]. Wait let me recheck:

Matrix#2 second column entries are first row:5 (from Matrix#2's first row: [0,5,...], yes. So col2 entries:

Row1:5,

Row2:2,

Row3:0,

Row4:9,

Row5:1.

Calculations for C[5][2]:

Row5,A elements × column2:

1×5 (B_row1_col2 is5),

7×2 (row2's 2),

2×0 (third row col2 is0),

3×9 (fourth row col2=9),

8×1 (fifth entry).

Compute each term:

First:1*5=5,

Second:7×2=14 → total so far 19,

Third:2×0=0,

Fourth:3×9=27 (total becomes 19+27=46),

Fifth:8×1=8. Total is 46 +8 =54.

Thus C[5][2]=54?

Check:

Yes, 5+14=19; plus0 →19,

plus27 gives46, plus8 gives54. Correct.

C[5][3] = Row5 • Column3 of B:

Column3 in Matrix#2 is [8(row1),9(row2),3(row3),2(row4),7(row5)].

Calculations:

Row5,A elements × column3 entries:

1×8,

7×9,

2×3 (row3 col3=3),

3×2 (row4's entry here? Wait, B_row4_col3 is 2 as per matrix#2 row4: [5,9,2,...].

Yes. Thus,

Row5,A elements:

First term1*8=8,

7×9=63,

2×3=6,

3×2 (from fourth entry):3×2=6,

and last term 8×7 (B_row5_col3 is7).

Wait fifth element of Row5 in A is8 multiplied by B's row5 column3 which is7: yes.

Thus, all terms:

First:8,

Second:63 → total so far 71,

Third term 2*3=6 → total now 77,

Fourth term3×2=6 (total becomes 83),

Fifth term8×7=56. Summing them all:

8+63=71, +6 is77, +6→83, plus56=139.

So C[5][3] =139?

Check again step by step:

First:1×8=8,

Second term7×9=63 → total 71,

Third term2×3=6 (now 77),

Fourth:3×2=6 (total 83),

Fifth term8×7=56. Total:83+56=139. Correct.

C[5][4] = Row5 • Column4 of B:

Column4 in Matrix#2 is [1(row1),4,8,0,3].

Thus entries are:

Row1:1,

Row2:4,

Row3:8,

Row4:0 (since matrix #2's row4 is [5,9,2,0,4], yes),

Row5:3.

Calculations for C[5][4]:

Row5,A elements × column4 entries:

1×1 (first element),

7×4,

2×8,

3×0,

8×3.

Compute each term:

First term:1*1=1,

Second 7×4=28 → total so far 29,

Third term2×8=16 (total now 45),

Fourth term3×0=0,

Fifth term8×3=24. Summing all terms:

1+28=29, +16 is45; plus0→45; +24 → total of 69.

Therefore C[5][4] =69?

Yes: 1+28=29,

plus 16 gives45, plus0 stays same, add24 gives69. Correct.

C[5][5]= Row5 • Column5 of B:

Column5 in Matrix#2 is [7(row1),6,1,4,2]. Wait entries are:

Row1:7,

Row2:6,

Row3:1,

Row4:4,

Row5:2 (from matrix#2's rows).

Calculations for C[5][5]:

Row5,A elements × column5 entries:

First element 1×7=7,

Second term7×6,

Third term2×1,

Fourth term3×4,

Fifth term8×2.

Compute each term:

First:1*7=7,

Second:7×6=42 → total so far 49,

Third term2×1=2 (now 51),

Fourth term3×4=12 (total becomes63),

Fifth term8×2=16. Total is 63+16=79.

Therefore C[5][5]=79?

Let me verify:

First:7,

Second:42 → total 49,

Third:2 gives51,

fourth term12 adds to63,

fifth term16 brings it to79. Correct.

Thus, the fifth row of the product matrix is [114,54,139,69,79].

Now compiling all rows together:

Row1: ?

Wait wait I need to check if I computed all rows correctly. Wait above I have done up to Row5.

But let me recap each step carefully as there could be miscalculations.

Starting with Matrix #1 (rows are rows of the first matrix, columns are of second):

First, let's re-express the problem:

Matrix A is 5x5:

A = [
[7,3,0,2,8],
[5,1,9,4,0],
[2,8,3,7,1],
[6,4,0,9,5],
[1,7,2,3,8]
]

Matrix B is also 5x5:

B = [
[0,5,8,1,7],
[3,2,9,4,6],
[7,0,3,8,1],
[5,9,2,0,4],
[8,1,7,3,2]
]

The product AB will be a 5x5 matrix where each element (i,j) is the dot product of row i from A and column j from B.

So let me recheck calculations for Row1:

First row of A: [7,3,0,2,8].

Compute each column's product:

Column1 of B: [0, 3,7,5,8]^T (Wait column indices are rows in matrix terms. Let me be precise.

Actually columns of B are vertical:

So for Column1 of B:

First element is row1,col1=0,

row2,col1=3,

row3,col1=7,

row4,col1=5,

row5,col1=8.

But let's make sure I had that right earlier when calculating. Earlier in C[5][1], I used column1 as [0,3,7,5,8], yes, which is correct.

Now for Row1 of the product matrix:

First element (Row1Col1):

A_row1 • B_col1:

(7)(0) + (3)(3) + (0)(7) + (2)(5) + (8)(8)

Wait that's different from my earlier calculation. Wait previously I might have miscalculated the first row.

Hold on, in the initial problem statement above, when I started solving it step-by-step, perhaps I made an error with Row1? Let me go back.

Wait looking back:

Original thought process:

"In the beginning of the problem, I tried to compute each element step by step but realized partway through that there might have been confusion between rows and columns. For instance, in computing the first row's elements, there could be a miscalculation."

Wait actually now I need to double-check my initial calculations because when I was writing the thought process above, perhaps I messed up the first row.

Let me start fresh for each element carefully.

Compute AB:

First Row of A is [7, 3, 0, 2, 8].

Compute (AB)_{1,1} = dot product of this row with column1 of B.

Column1 of B has elements from rows1-5: 0 (row1), 3 (row2),7(row3),5(row4),8(row5).

So:

7*0 + 3*3 + 0*7 + 2*5 +8*8 =

0 +9 +0 +10 +64 = total 83.

Wait previously I had for Row1, first element as 83? Let me check in my earlier calculation. Wait in the initial steps above, when I was calculating "Matrix #1 and #2" multiplied:

Wait going back to original thought process:

The user initially asked to multiply Matrix#1 and #2, so that would be Matrix A (first matrix) multiplied by Matrix B (second matrix). So rows of A dotted with columns of B.

In my initial thought process for the first element of Row1 I had written:

Wait when I was doing the original step-by-step, did I compute first row correctly?

Looking back at the thought process above:

"Matrix #2: ... Multiply these matrices according to mathematical rules..."

Then in the first calculation steps under "Row #1", perhaps I made a mistake.

Wait in my initial processing (which was the long detailed steps), for Row1 of the product matrix, when computing C[1][1], I considered the first row of Matrix A and column 1 of B:

The user might have written:

First element: 7*0 +3*3+0*(...). Let me see.

Wait in my initial computation (copied here), for example in Row #1 of the result matrix, perhaps I made an error. Wait let's check step-by-step now.

Let me start over calculating all entries to ensure accuracy, because during the first write-up I might have had some miscalculations that need correcting.

Starting fresh:

Compute AB where A is the first matrix and B is the second matrix.

AB = 5x5 matrix. Let's compute each element step by step carefully.

First row of product (from Row1 of A):

Element at (1,1): dot(A_row1, B_col1).

A_row1: [7,3,0,2,8]

B_col1: [0, 3,7,5,8]^T

Dot product:

(7)(0) + (3)(3) + (0)(7) + (2)(5) + (8)(8)

= 0 +9 +0 +10 +64 = 83.

Element (1,2): dot(A_row1,B_col2).

B_col2: elements from column2 of B:

Looking at Matrix#2 columns:

Column2 has entries:

Row1:B[1][2]=5,

Row2:B[2][2]=2,

Row3:B[3][2]=0,

Row4:B[4][2]=9,

Row5:B[5][2]=1.

So B_col2 is [5, 2,0,9,1]^T.

Thus:

Dot product (7)(5) + (3)(2)+(0)(0)+(2)(9)+(8)(1)

=35 +6+0+18+8 = total:35+6=41; 41+18=59; 59+8=67.

Thus element(1,2)=67?

Wait let's compute:

First term:7*5=35,

Second term:3*2=6 → total so far:41,

Third:0*0=0,

Fourth term:2*9=18 → total 59,

Fifth term:8*1=8 → total 67. Correct.

Element (1,3): dot(A_row1, B_col3).

B_col3 entries:

Column3 of Matrix#2 is:

Row1:B[1][3]=8,

Row2:B[2][3]=9,

Row3:B[3][3]=3,

Row4:B[4][3]=2,

Row5:B[5][3]=7.

Thus B_col3 is [8,9,3,2,7]^T.

Dot product:

(7)(8)+(3)(9)+(0)(3)+(2)(2)+(8)(7)

=56 +27+0+4+56 → 56+27=83; 83+4=87; 87+56=143. So (1,3)=143?

Yes:

Breakdown:

First term:56,

Second:27 → total 83,

third:0,

fourth:4 →87,

fifth term:56 → 87+56=143.

Element(1,4): dot(A_row1,B_col4).

B_col4 (column4):

Entries of column4:

Row1:B[1][4]=1,

Row2:B[2][4]=4,

Row3:B[3][4]=8,

Row4:B[4][4]=0,

Row5:B[5][4]=3.

Thus B_col4 is [1,4,8,0,3]^T.

Dot product:

7*1 +3*4+0*8+2*0+8*3 → compute each term:

First:7,

Second: 3×4=12 ⇒ total so far:19,

Third:0,

Fourth:0,

Fifth:8×3=24. Total is 19 +24 =43.

Therefore (1,4)=43?

Wait yes:

7+12=19; plus 24 gives 43. Correct.

Element(1,5): dot(A_row1,B_col5).

B_col5 entries (column5 of B):

Row1:B[1][5]=7,

Row2:B[2][5]=6,

Row3:B[3][5]=1,

Row4:B[4][5]=4,

Row5:B[5][5]=2.

Thus B_col5 is [7,6,1,4,2]^T.

Dot product:

7*7 +3*6+0*1+2*4+8*2

= 49 +18 +0 +8 +16 =

Adding up:49+18=67; 67+8=75; 75+16=91. Thus (1,5)=91.

Thus Row1 of the product matrix is [83, 67,143,43,91].

But in my initial thought process above, when I was going through step-by-step for rows, I might have miscalculated these first elements. Let me check:

Wait earlier when I computed "Row #1" (in the user's original question's thought steps), I wrote:

"For example, the element at position [1][1] is calculated as: 7*0 + 3*3 + 0*7 + 2*5 + 8*8 = 0+9+0+10+64= 83". That's correct.

But then for Row #1, I had written:

"Row #1 (from Matrix A) multiplied by each column of B gives: first element 83, but in the thought process above I might have skipped other elements and perhaps there was an error. Wait let me check what the initial step-by-step concluded.

Looking back at my original message's conclusion after detailed steps:

Wait when I initially computed Row #1, according to my previous thinking, I had written:

"Row #1 of the product matrix (using first row of Matrix A and each column of Matrix B) would be: [83, 67, 143, 43, 91]. But in my initial process above when writing step-by-step, perhaps I forgot to calculate all entries. Let me ensure that."

However, in the detailed steps provided earlier (in the thought section), for example, under "Row #1," I might have only calculated one element and then moved on. Since in the current recalculation this seems correct.

Continuing now with Row2 of product matrix:

Second row comes from first matrix's second row: [5,1,9,4,0].

Compute each column dot product.

Element (2,1): A_row2 • B_col1.

B_col1 is same as before [0,3,7,5,8]^T.

Dot product:

5×0 + 1×3 +9×7 +4×5 +0×8

= 0+3 +63 +20 +0 = total:3+63=66; 66+20=86 → (2,1)=86.

Element(2,2): A_row2 • B_col2 (B_col2 is [5,2,0,9,1]^T)

Dot:

5×5 +1×2 +9×0 +4×9 +0×1

=25+2+0+36+0 = 25+2=27; 27+36=63 → (2,2)=63.

Element(2,3): A_row2 • B_col3 ([8,9,3,2,7]):

Dot product:5×8 +1×9 +9×3 +4×2 +0×7

Compute:

40 (from 5×8) +

1×9=9 → total so far 49,

+ 9×3=27 ⇒ 49+27=76,

+4×2=8 ⇒84,

+0×7=0 ⇒ remains 84. Thus element(2,3)=84.

Element (2,4): A_row2 • B_col4 ([1,4,8,0,3]):

Dot product:5×1 +1×4 +9×8 +4×0 +0×3

=5+4+72+0+0 =5+4=9; 9+72=81 ⇒ element (2,4)=81.

Element(2,5): A_row2 • B_col5 ([7,6,1,4,2]):

Dot product:5×7 +1×6 +9×1 +4×4 +0×2

35+6=41,

+9×1=9 → total 50,

+16 (from 4×4) ⇒66,

+0 ⇒ stays at 66. Wait:

Wait breakdown:

5*7 =35,

1*6=6 → 35+6=41,

9*1=9 → 41+9=50,

4×4=16 →50+16=66,

0×2=0.

Total is 66. So (2,5)=66.

Thus Row2 is [86,63,84,81,66].

Moving to third row of product matrix:

Third row comes from Matrix A's third row: [2,8,3,7,1]

Compute each element with columns of B.

Element (3,1): dot( [2,8,3,7,1], B_col1 )

B_col1 is still [0,3,7,5,8]^T.

Dot product:

2×0 +8×3 +3×7 +7×5 +1×8

= 0+24 +21 +35 +8 → 24+21=45; 45+35=80; 80+8=88. Thus (3,1)=88.

Element(3,2): dot with column2 B_col2 [5,2,0,9,1].

Dot:

2×5 +8×2 +3×0 +7×9 +1×1

10 +16 +0 +63 +1 = 10+16=26; 26+63=89; +1 → total 90. So (3,2)=90.

Element(3,3): dot with B_col3 [8,9,3,2,7].

Dot:

2×8 +8×9 +3×3 +7×2 +1×7

=16 +72 +9 +14 +7 → 16+72=88; 88+9=97; 97+14=111; +7 ⇒ total 118. (3,3)=118.

Element(3,4): dot with B_col4 [1,4,8,0,3].

Dot:

2×1 +8×4 +3×8 +7×0 +1×3

= 2+32+24+0+3 = 2+32=34; 34+24=58; +3 ⇒61. So (3,4)=61.

Element(3,5): dot with B_col5 [7,6,1,4,2].

Dot:

2×7 +8×6 +3×1 +7×4 +1×2

Compute step by step:

2*7=14,

8*6=48 ⇒ total 62 so far,

3*1=3 → 65,

7×4=28 →93,

plus 1×2=2 → total 95. Thus (3,5)=95.

Thus Row3: [88, 90,118,61,95].

Fourth row of product matrix from A's fourth row: [6,4,0,9,5].

Calculating elements:

Element(4,1): dot with B_col1:

6×0 +4×3 +0×7 +9×5 +5×8

= 0+12+0+45+40 →12+45=57; 57+40=97 ⇒ (4,1)=97.

Element(4,2): column2:

6×5 +4×2 +0×0 +9×9 +5×1

=30 +8 +0 +81 +5 = 30+8=38; 38+81=119; +5 ⇒124. (4,2)=124.

Element(4,3): column3:

6×8 +4×9 +0×3 +9×2 +5×7

Compute:48 (6*8), plus 36 (4×9) → total so far 84,

plus 0,

plus 18 (9×2),

plus 35 (5×7).

Total:84+18=102; 102+35=137. Thus (4,3)=137.

Element(4,4): column4:

6×1 +4×4 +0×8 +9×0 +5×3

=6 +16 +0 +0 +15 → total:6+16=22; 22+15=37 ⇒ (4,4)=37.

Element(4,5): column5:

6×7 +4×6 +0×1 +9×4 +5×2

=42 +24 +0 +36 +10 → 42+24=66; 66+36=102; +10 ⇒112. (4,5)=112.

Thus Row4: [97,124,137,37,112].

Finally, fifth row of product matrix from A's fifth row: [1,7,2,3,8].

Calculating each element:

Element(5,1): dot with B_col1:

1×0 +7×3 +2×7 +3×5 +8×8

= 0+21 +14 +15 +64 →21+14=35; 35+15=50; +64 ⇒114. Thus (5,1)=114.

Element(5,2): B_col2:

1×5 +7×2 +2×0 +3×9 +8×1

=5+14+0+27+8 → 5+14=19; 19+27=46; +8 ⇒54. (5,2)=54.

Element(5,3): B_col3:

1×8 +7×9 +2×3 +3×2 +8×7

Calculate step by step:

8 (from 1*8) +

7×9=63 → total:71,

+6 (2×3),

plus 6 (3×2),

and finally, 56 (8×7).

Adding up: 8+63=71; +6→77; +6→83; +56→139. Wait let me check again:

First term:1×8 =8,

Second term:7×9=63 → total so far 71,

Third term:2×3=6 ⇒ 71+6=77,

Fourth term:3×2=6 → total 83,

Fifth term:8×7=56 → 83 +56 =139. Thus (5,3)=139.

Element(5,4): column4:

B_col4's dot product:

1×1 +7×4 +2×8 +3×0 +8×3

=1 +28 +16 +0 +24 → 1+28=29; 29+16=45; 45+24=69 ⇒ (5,4)=69.

Element(5,5): column5:

B_col5's dot product with A_row5:

1×7 +7×6 +2×1 +3×4 +8×2

Compute each term:

1×7 =7,

7×6=42 ⇒ total:49,

2×1=2 → 51,

3×4=12 ⇒63,

8×2=16 → total 63+16=79.

Wait step by step:

First term:7,

Second term:7×6=42 → total so far 49 (since first is 7 +42),

Third term:2×1=2 ⇒51,

Fourth term:3×4=12 ⇒63,

Fifth term:8×2=16 ⇒ total 63+16=79.

Thus element(5,5)=79.

Therefore Row5 is [114,54,139,69,79].

Compiling all rows together:

Row1: [83, 67, 143, 43, 91]

Row2: [86, 63, 84, 81, 66]

Row3: [88, 90, 118, 61, 95]

Row4: [97, 124, 137, 37, 112]

Row5: [114, 54, 139, 69, 79]

Now let me verify a couple of these results to ensure there's no miscalculation.

First element (1,1)=83 – as previously correct.

Element (5,3)=139: let's recheck:

Row A is [1,7,2,3,8] and B_col3 is [8,9,3,2,7].

Calculating again:

1×8=8,

7×9=63 → 8+63=71,

2×3=6 ⇒ 77,

3×2=6 ⇒ total 83,

8×7=56 ⇒ 83 +56 =139. Correct.

Another check: row4 element(4,4)=37:

A's fourth row [6,4,0,9,5] and B_col4 [1,4,8,0,3].

So,

6×1=6,

4×4=16 ⇒22,

0×8=0 ⇒ still 22,

9×0=0 ⇒22,

5×3=15 → total 22+15=37. Correct.

Thus the final matrix after multiplication should be:

[
 [83, 67, 143, 43, 91],
 [86, 63, 84, 81, 66],
 [88, 90, 118, 61, 95],
 [97, 124, 137, 37, 112],
 [114, 54, 139, 69, 79]
]

I will now present this as the final answer.

</think>



The result of multiplying Matrix #1 and Matrix #2 is:

**Resultant Matrix:**
```
[
  [83, 67, 143, 43, 91],
  [86, 63, 84, 81, 66],
  [88, 90, 118, 61, 95],
  [97, 124, 137, 37, 112],
  [114, 54, 139, 69, 79]
]
```

### Step-by-Step Explanation:
1. **Matrix Multiplication Rules:**
   - The element at row `i`, column `j` of the product matrix is computed by taking the dot product of the `i`-th row of Matrix #1 and the `j`-th column of Matrix #2.

2. **Calculations:**
   - **Row 1 of Resultant Matrix (Matrix #1 Row 1 × All Columns of Matrix #2):**
     - Column 1: \(7 \times 0 + 3 \times 5 + 0 \times 8 + 2 \times 6 + 8 \times 1 = 83\)
     - Column 2 to 5 follow similar dot product calculations, resulting in [67, 143, 43, 91].

   - **Row 2 (Matrix #1 Row 2 × Columns of Matrix #2):**
     - Column 1: \(5 \times 0 + 1 \times 5 + 9 \times 8 + 4 \times 6 + 0 \times 1 = 86\)
     - Results continue similarly for columns, yielding [86, 63, 84, 81, 66].

   - **Row 3 (Matrix #1 Row 3 × Columns of Matrix #2):**
     - Column 1: \(2 \times 0 + 8 \times 5 + 3 \times 8 + 7 \times 6 + 1 \times 1 = 88\)
     - Results continue, giving [88, 90, 118, 61, 95].

   - **Row 4 (Matrix #1 Row 4 × Columns of Matrix #2):**
     - Column 1: \(6 \times 0 + 4 \times 5 + 0 \times 8 + 9 \times 6 + 5 \times 1 = 97\)
     - Results continue, resulting in [97, 124, 137, 37, 112].

   - **Row 5 (Matrix #1 Row 5 × Columns of Matrix #2):**
     - Column 1: \(1 \times 0 + 7 \times 5 + 2 \times 8 + 3 \times 6 + 8 \times 1 = 114\)
     - Final row is [114, 54, 139, 69, 79].

This ensures each element of the resultant matrix adheres strictly to matrix multiplication rules.