Orders of a unit in a matrix ring

We calculate the order and factored order of a random matrix over a finite field.

> M := MatrixAlgebra(GF(23), 20);
> A := Random(M);
> A;
[ 6  1 14  6 11 21 16  7  7 15  3 21  7  5 21 17  7  6 16  1]
[15 17  5  4 13  5 18 21 11 12 18  1  7 22 12  1 19 22  9 15]
[21 13  4 13 17  0  0 21  3  0 13 13  0  1 17  9  9 18 11 21]
[14  5 20 20 22  4  5 21  0  3  7 14  1  0 12 19  5 18  0 20]
[19  9 14 19  4  5 20  8 22  8 12  9  9 22  9 16 10 14  8  5]
[21 11 12  6 11 19 12  9  8  9  7  0 10  0  3 16 21  2 19  9]
[14  9  9 16 22  5  0 14  6  4  2 11 20 17  7 10  7  7 13 10]
[22 21  2 22 11 18  7  3 19  7  2 18 11  3 10 18 10  8  1 19]
[ 5 17 10 17  1 22  8  3 19 13 22 20  8 12 17 14  3 15 12  4]
[ 0 16  6  7 19 19 10  3 15 21  3 22 13 19 22  6 19  1 12 12]
[18 19 18  0 15  5 19 22  6  9 22 20 16 17 12  2  5  2 22 16]
[21  1 22  6 18 14  2  7  8 15  9 20 11 15 20  7 16  3  5  8]
[ 8 19 18  3  7  5  7 19 22 13  4 13  7  4 11 21  3 14  8  3]
[ 5 11 15 15 19  0  1 12  0  8  0  1 18 10  8  0  5  0 15 11]
[21  7 18  2  5 22 21  8  6  5 18 17 22 15 12 13  2  7  6  4]
[ 3  7 13 12 19  3 10 16 18 20 10 21 11 21  2 19 11  6 13  8]
[15 20 18 15 12  7 18  2  3 16 18  4  7 14 17 16  0 22 15  1]
[15 11 21 12  9  2  0 12 12 21 12 10 11 20  8  2 10 17 13 21]
[15 10 17  0 19 13 21 21 16 10 13  0 10 12 13 21  3 10 20  7]
[ 3 21 11 12 16 13  2 17 21 12 16 11 14  9  7 10 19 10  0  7]
>
> Order(A);
216138319375440
> FactoredOrder(A);
[ <2, 4>, <3, 1>, <5, 1>, <7, 1>, <11, 1>, <19, 1>, <79, 1>, <7792003, 1> ]

Previous Group: Matrix algebra over a polynomial ring