AFC

mat_orig <- matrix(c(
1,  1,  1,  0,  0,
0,  4,  8,  5,  0,
0,  0,  0,  3,  4
), ncol = 3)

colnames(mat_orig) <- c('[5,10[','[10,15[','[15,20]')

rownames(mat_orig) <- c('[0,4[','[4,8[','[8,12[','[12,16[','[16,20[')

mat_orig


mat_fi <- matrix(c(
0.888888889,    0.088888889,    0, -0.111111111,    -0.111111111,
-0.62962963,    0.17037037, 0.259259259,    -0.00462963,    -0.62962963,
-0.259259259, -0.259259259, -0.259259259,   0.115740741,    0.740740741
), ncol = 3)

mat_fi


mat_mp <- matrix(c(
 9.00000, 0, 0,
 0, 1.58824, 0,
 0, 0, 3.85714
), ncol=3)

mat_mp

mat_mn <- matrix(c(
 27.0000, 0, 0, 0, 0,
 0, 5.4000, 0, 0, 0,
 0, 0, 3.0000, 0, 0,
 0, 0, 0, 3.3750, 0,
 0, 0, 0, 0, 6.7500
), ncol=5)

mat_mn


res_col <- eigen(mat_mp %*% t(mat_fi) %*% solve(mat_mn) %*% mat_fi)

res_col

comp_col <- matrix(c(
    mat_fi %*% matrix(res_col$vectors[,1], ncol=1) * -2.0897 ,
    mat_fi %*% matrix(res_col$vectors[,2], ncol=1) *  4.75
), ncol=2)


comp_col


library(FactoMineR)

test_ca <- CA(mat_orig)
comp_col
test_ca$row$coord