So I cant find the answer of my assignment in google about this matrix question

Question

Assume A is 3 by 3 matrix. Show that (A^T )^-1= A by using your own example. Can anyone give me the example

Answer 1

This only works if A is an orthogonal matrix.

For example

A = .707 - .707 0

.707 .707 0

0 0 1

However it does not work is A is not orthogonal

For example

A = 2 0 0

0 2 0

0 0 2

The inverse is

A^-1 = .5 0 0

0 .5 0

0 0 .5

which is not the transpose of A.