A matrix A is orthogonal if itstranspose is equal to it inverse. So

AT is the transpose of A and A-1 is the inverse.

We have AT=A-1

So we have :

AAT= I, the identity matrix

Since it is MUCH easier to find a transpose than an inverse, these matrices are easy to compute with. Furthermore, rotation matrices are orthogonal.

The inverse of an orthogonal matrix is also orthogonal which can be easily proved directly from the definition.

For the matrix , verify that

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.

It need not be, so the question makes no sense!

the transpose of null space of A is equal to orthogonal complement of A