So i got this question on my assignment about matrix it says that i have to proof one of the matrix properties

Question

Assume A is a 3 x 3 matrix with |A| ≠ 0. Show that A^-1XA = I , by using your own example.

I've tried using some random numbers but it seems that it doesn't work, so there must be some rules to proof this matrix properties

Answer 1

An easy example is to let A be a 3x3 diagonal matrix with nonzero entries on the main diagonal. Say for example (if a_ij is the entry in row i column j) let a₁₁ = 2, a₂₂ = 3, a₃₃ = 4 and set all other entries equal to 0. Then A^-1 is the 3x3 diagonal matrix with 1/2, 1/3, 1/4 as the entries on the main diagonal and has all other entries equal to zero. It's easy to check that AA^-1 is the 3x3 identity matrix.