Can someone help me find the inverse of the 2x2 matrix I43I I21I in GL(2,Z_5)?

Answer 1

Let M be your matrix.

det M = 4 * 1 - 2 * 3 = 4 - 1 = 3

M^(-1) = (det M)^(-1) M_c^T

where M_c is the cofactor matrix and M_c^T is its transpose. which is |1 -3| |-2 4| = |1 2| |3 4|.

Now, det M ^(-1) = 3^(-1) = 2 in Z_5

So the matrix inverse is

| 2 4 |
| 1 3 |

Answer 2

4 3
2 1
In the following all calculations are done modulo 5.
The determinant of the matrix in Z_5 is 4 - 6 = 3
So the inverse is
1/3* 1 -3
.......-2 4

= 2* 1 2
........3 4
= 2 4
...1 3

Check:
In Z_5
4 3 times.... 2 4
2 1 ............ 1 3
equals
1 0
0 1.

Answer 3

The rule for inverting a 2x2 matrix: [a b] [c d] is 1 / (ad-bc) times [d -b] [-c a]