Iteration of 3  equations  step  \[CapitalDelta]t = 0.01

\[DifferentialD]^2Subscript[q, j + 1]/\[DifferentialD]t^2 = 
  Subscript[a, 1] (Subscript[q, j + 1] - Subscript[q, j]) - 
   Subscript[a, 3] \[DifferentialD]Subscript[q, j]/\[DifferentialD]t -
    Subscript[a, 
    5] \[DifferentialD]^2Subscript[q, j]/\[DifferentialD]t^2;
\[DifferentialD]Subscript[q, j + 1]/\[DifferentialD]t = 
  Subscript[a, 2] (Subscript[q, j + 1] - Subscript[q, j]) + 
   Subscript[a, 4] \[DifferentialD]Subscript[q, j]/\[DifferentialD]t +
    Subscript[a, 
    6] \[DifferentialD]^2Subscript[q, j]/\[DifferentialD]t^2;
(Subscript[a, 1] M0 + Subscript[a, 2] C0 + K0) (Subscript[q, 
    j + 1]) = (Subscript[f, 
    j + 1]) + (Subscript[a, 1] M0 + Subscript[a, 2] C0) Subscript[q, 
    j] + (Subscript[a, 3] M0 - 
      Subscript[a, 4] C0) \[DifferentialD]Subscript[q, 
     j]/\[DifferentialD]t + (Subscript[a, 5] M0 - 
      Subscript[a, 6] C0) \[DifferentialD]^2Subscript[q, 
     j]/\[DifferentialD]t^2;

where

Subscript[a, 1] = 1/(\[Beta]*\[CapitalDelta]t^2);
Subscript[a, 2] = \[Gamma]/(\[Beta]*\[CapitalDelta]t);
Subscript[a, 3] = 1/(\[Beta]*\[CapitalDelta]t);
Subscript[a, 4] = 1 - \[Gamma]/\[Beta];
Subscript[a, 5] = 1/(2*\[Beta]) - 1;
Subscript[a, 6] = \[CapitalDelta]t*(1 - \[Gamma]/(2*\[Beta]));
\[Gamma] = 1/2;
\[Beta] = 1/6;
M0 = \!\(\*
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FractionBox["13427", "72187500"], "0", "0", "0", "0", "0", "0", "0", 
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FractionBox["13427", "32812500"], "0", 
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FractionBox["13427", "360937500"], "0", 
RowBox[{"-", 
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RowBox[{"-", 
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FractionBox["13427", "1564062500"], "0", "0", "0", "0", "0"},
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FractionBox["13427", "4692187500"], "0", "0", "0", "0"},
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FractionBox["13427", "98437500000000"], "0", 
RowBox[{"-", 
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RowBox[{"-", 
FractionBox["13427", "196875000000000"]}], "0", 
FractionBox["13427", "216562500000000"], "0"},
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RowBox[{"-", 
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FractionBox["13427", "402187500000000"]}
},
GridBoxAlignment->{
        "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
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GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.7]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}}], "", ")"}],
Function[BoxForm`e$, 
MatrixForm[BoxForm`e$]]]\);

K0 = \!\(\*
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FractionBox["560000000", "87"], "0", "0", "0", "0", 
FractionBox[
RowBox[{"6400000000", " ", "q10"}], "83259"], "0", 
RowBox[{"-", 
FractionBox[
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         "0"},
{"0", 
FractionBox["112000000", "29"], "0", "0", "0", 
FractionBox[
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FractionBox["80000000", "29"], "0", "0", 
FractionBox[
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RowBox[{"-", 
FractionBox[
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{"0", "0", "0", 
FractionBox["560000000", "261"], "0", 
FractionBox[
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RowBox[{"-", 
FractionBox[
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         "0"},
{"0", 
RowBox[{"-", 
FractionBox[
RowBox[{"25600000000", " ", "q10"}], "360789"]}], "0", 
RowBox[{"-", 
FractionBox[
RowBox[{"1510400000000", " ", "q10"}], "6133413"]}], 
FractionBox["2000000000", "17487"], 
RowBox[{"-", 
FractionBox[
RowBox[{"2560000000000", " ", "q10"}], "59289659"]}], 
RowBox[{"-", 
FractionBox["1000000000", "52461"]}], 
FractionBox[
RowBox[{"143680000000000", " ", "q10"}], "3379510563"], 
FractionBox["8096804013267", "244118641"], "0", 
RowBox[{"-", 
FractionBox["3685874671089", "666774728"]}], "0"},
{"0", 
RowBox[{"-", 
FractionBox[
RowBox[{"25600000000", " ", "q20"}], "360789"]}], "0", 
RowBox[{"-", 
FractionBox[
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RowBox[{
FractionBox["2000000000", "52461"], "-", 
FractionBox[
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RowBox[{
RowBox[{"-", 
FractionBox["7106725169", "820217"]}], "+", 
FractionBox[
RowBox[{"143680000000000", " ", "q20"}], "3379510563"]}], "0", 
FractionBox["3685874671089", "333387364"], "0", 
RowBox[{"-", 
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{"0", 
RowBox[{"-", 
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RowBox[{"-", 
FractionBox[
RowBox[{"1510400000000", " ", "q30"}], "6133413"]}], 
RowBox[{"-", 
FractionBox["1000000000", "52461"]}], 
RowBox[{"-", 
FractionBox[
RowBox[{"2560000000000", " ", "q30"}], "59289659"]}], 
FractionBox["12106725169", "698644"], 
FractionBox[
RowBox[{"143680000000000", " ", "q30"}], "3379510563"], 
RowBox[{"-", 
FractionBox["3685874671089", "666774728"]}], "0", 
FractionBox["100000000", "19899"], "0"},
{"0", 
RowBox[{"-", 
FractionBox[
RowBox[{"25600000000", " ", "q40"}], "360789"]}], "0", 
RowBox[{"-", 
FractionBox[
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RowBox[{
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FractionBox["7106725169", "820217"]}], "-", 
FractionBox[
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RowBox[{
FractionBox["8785650026", "941561"], "+", 
FractionBox[
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RowBox[{"-", 
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FractionBox["27308074623", "10091777"]},
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FractionBox["8096804013267", "244118641"], "0", 
RowBox[{"-", 
FractionBox["3685874671089", "666774728"]}], "0", 
FractionBox["248364949983", "25817935"], "0", 
RowBox[{"-", 
FractionBox["2900000", "1809"]}], "0"},
{"0", "0", "0", "0", "0", 
FractionBox["3685874671089", "333387364"], "0", 
RowBox[{"-", 
FractionBox["50000000", "19899"]}], "0", 
FractionBox["168248240", "52461"], "0", 
RowBox[{"-", 
FractionBox["14500000", "19899"]}]},
{"0", "0", "0", "0", 
RowBox[{"-", 
FractionBox["3685874671089", "666774728"]}], "0", 
FractionBox["100000000", "19899"], "0", 
RowBox[{"-", 
FractionBox["2900000", "1809"]}], "0", 
FractionBox["841412720", "577071"], "0"},
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RowBox[{"-", 
FractionBox["50000000", "19899"]}], "0", 
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FractionBox["14500000", "19899"]}], "0", 
FractionBox["2653448907", "3378824"]}
},
GridBoxAlignment->{
        "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
         "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.7]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}}], "", ")"}],
Function[BoxForm`e$, 
MatrixForm[BoxForm`e$]]]\);
C0 = r*K0;
r = 0.01;


From initial conditions I determined 

q0 = \!\(\*
TagBox[
RowBox[{"(", "", GridBox[{
{
FractionBox["7425000", "3241"]},
{"0"},
{
FractionBox["21450000", "3241"]},
{"0"}
},
GridBoxAlignment->{
        "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
         "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.7]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}}], "", ")"}],
Function[BoxForm`e$, 
MatrixForm[BoxForm`e$]]]\);
or in this form where we can see in K0 matrix
\!\(\*
TagBox[
RowBox[{
RowBox[{
RowBox[{"q10", ":=", 
FractionBox["7425000", "3241"]}], ";"}], "\[IndentingNewLine]", 
    "\[IndentingNewLine]", 
RowBox[{
RowBox[{"q20", ":=", "0"}], ";"}], "\[IndentingNewLine]", 
    "\[IndentingNewLine]", 
RowBox[{
RowBox[{"q30", ":=", 
FractionBox["21450000", "3241"]}], ";"}], "\[IndentingNewLine]", 
    "\[IndentingNewLine]", 
RowBox[{
RowBox[{"q40", ":=", "0"}], ";"}], "\[IndentingNewLine]", 
    "\[IndentingNewLine]", "\[IndentingNewLine]"}],
Function[BoxForm`e$, 
MatrixForm[BoxForm`e$]]]\)