Solution of matrix equations

We construct a 10 by 10 matrix X of integers.

> M := MatrixRing(IntegerRing(), 10);
> M;
Full Matrix Algebra of degree 10 over Integer Ring
> X := M ! [i+2*j: i,j in [1..10]];
> X;
[ 3  5  7  9 11 13 15 17 19 21]
[ 4  6  8 10 12 14 16 18 20 22]
[ 5  7  9 11 13 15 17 19 21 23]
[ 6  8 10 12 14 16 18 20 22 24]
[ 7  9 11 13 15 17 19 21 23 25]
[ 8 10 12 14 16 18 20 22 24 26]
[ 9 11 13 15 17 19 21 23 25 27]
[10 12 14 16 18 20 22 24 26 28]
[11 13 15 17 19 21 23 25 27 29]
[12 14 16 18 20 22 24 26 28 30]

We set V to be the length 10 vector (5, 4, 3, 5, 6, 4, 3, 7, 6, 5) and then set W to be VX.

> V := RSpace(IntegerRing(), 10) ! [5, 4, 3, 5, 6, 4, 3, 7, 6, 5];
> V;
(5 4 3 5 6 4 3 7 6 5)
> W := V * X;
> W;
( 373  469  565  661  757  853  949 1045 1141 1237)

Now we find a solution S of the equation SX = W together with the kernel K of X.

> S, K := Solution(X, W);      
> S;
(-181  229    0    0    0    0    0    0    0    0)
> K;
RSpace of degree 10, dimension 8 over Integer Ring
Echelonized basis:
( 1  0  0  0  0  0  0  0 -9  8)
( 0  1  0  0  0  0  0  0 -8  7)
( 0  0  1  0  0  0  0  0 -7  6)
( 0  0  0  1  0  0  0  0 -6  5)
( 0  0  0  0  1  0  0  0 -5  4)
( 0  0  0  0  0  1  0  0 -4  3)
( 0  0  0  0  0  0  1  0 -3  2)
( 0  0  0  0  0  0  0  1 -2  1)

Finally we note that V - S is in the kernel as expected.

> V - s in K;
true

Next Group: Smith form of integer matrices Previous Group: Z-modules