If A is an orthogonal matrix then why is it's inverse also orthogonal?

Answer 1

First let's be clear on the definitions.
A matrix M is orthogonal if MT=M-1
Or multiply both sides by M and you have
1) M MT=I
or
2) MTM=I

Where I is the identity matrix.

So our definition tells us a matrix is orthogonal if its transpose equals its inverse or if the product ( left or right) of the the matrix and its transpose is the identity.

Now we want to show why the inverse of an orthogonal matrix is also orthogonal.

Let A be orthogonal. We are assuming it is square since it has an inverse.

Now we want to show that A-1 is orthogonal.
We need to show that the inverse is equal to the transpose.
Since A is orthogonal, A=AT
Let's multiply both sides by A-1

A-1 A= A-1 AT
Or A-1 AT =I
Compare this to the definition above in 1) (M MT=I)
do you see how A-1 now fits the definition of orthogonal?
Or course we could have multiplied on the left and then we would have arrived at 2) above.