1. ## Matrix Inverse Proof

Show that $(A ^{-1})^T = (A ^T)^{-1}$

Assuming A is invertible.

2. If you can take the assumption that the determinant of a matrix and its transpose are equal, you can do it by induction and the definition of the inverse as it is defined in terms of cofactors.

Here's Wikipedia:

Invertible matrix - Wikipedia, the free encyclopedia

... it might help.

3. $A A^{-1} = I$

$(A A^{-1})^T = (I)^T$

$(A^{-1})^T A^T = I$

By definition of inverse matrix, $(A^{-1})^T$ is the inverse of $A^T$, hence:

$(A^{-1})^T = (A^T)^{-1}$