a11 = -2; a12 = -3; a21 = 3; a22 = 12
We want to find the inverse A^(-1) such as A*A^(-1) = I, where I is the matrix identity.
Let a'11, a12', a21' and a22' be the terms of the matrix A^(-1).
In this case, we have:
I) a11*a11' + a12*a21' = 1
II) a11*a12' + a12*a22' = 0
III) a21*a11' + a22*a21' = 0
IV) a21*a12' + a22*a22' = 1
I) -2a11' - 3a21' = 1
II) -2a12' - 3a22' = 0
III) 3a11' + 12a21' = 0
IV) 3a12' + 12a22' = 1
Adding 4 times equation I to equation III, we have:
-5a11' = 4
a11' = -4/5
Applying this result in equation I, we have:
8/5 -3a21' = 1
-3a21' = -3/5
a21' = 1/5
Adding 4 times equation II to equation IV, we have:
-5a12' = 1
a12' = -1/5
Applying this result in equation II, we have:
2/5 - 3a22' = 1
-3a22' = 3/5
a22' = -1/5
Therefore, the matrix A^(-1) is given by [a11' a12' a21' a22'] = [-4/5 -1/5 1/5 -1/5]