(b) Since A ∼ B, there exists an invertible matrix P such that B = P−1AP. ... (a) Explain why the inverse of a permutation matrix equals its transpose: ...
Since interchanging two rows is a self-reverse operation, every elementary permutation matrix is invertible and agrees with its inverse, P = P−1 or P2 = I.
The transpose of a permutation matrix is its inverse. ... This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here ...
Inverse of a permutation matrix is its transpose. This is because permutation matrices are orthogonal. Intuitively this makes sense because when you permute a matrix, the rows/columns you swap can be obtained back by applying the reverse of the same operation.