Suppose that A and B are 3 times 3 invertible matrices such

Suppose that A and B are 3 times 3 invertible matrices such that A^2 = A and B^T = B^-1. Then by correctly using the given information and properties of matrices show A^-1 B(A^T B)^T = I_3.

Solution

Given: A²=A and B^T=B^-1

To prove: A^-1B(A^TB)^T=I₃

L.H.S:

A^-1B(A^TB)^T= A^-1B(B^T(A^T)^T) [because (AB)^T=B^TA^T]

= A^-1BB^-1A [because B^T=B^-1 and (A^T)^T=A]

= A^-1I₃A [because BB^-1=I_n where n is the order of matrix B]

= A^-1A [because I₃A=A]

= I₃= R.H.S

HENCE PROVED