If you know how to find a matrix inverse using gaussian elimination, this is going to be a piece of cake! For

, you perform gaussian elimination on the matrix but instead of dividing you will multiply by the inverse of the element in question. Then after you perform calculations you will just mod out the modulus.

Let A be your matrix with entries in

:

The determinant of this matrix is just

and is thus non-zero. Our matrix A is thus invertible.

Set it up as for Gaussian elimination

The inverse of 4 in

is 4 itself and the inverse of 3 is 2 so we can multiply each row by each of these respectively to get

Add -1 times the first row to the second to get

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From here on you multiply the 2nd row by the inverse of 2, which is 3. Then subtract 4 times the second row times from the first

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And that should be your inverse!(unless i've made a technical error)

EDIT: Lets perform a check!