# Showing a matrix is invertible

## Homework Statement

let A and B be nXn invertible matrices. Show that (A.B)T is invertible by finding it's inverse

## The Attempt at a Solution

(A.B)T=BT.AT

Last edited:

CAF123
Gold Member
You are looking for a matrix ##C## such that ##(AB)^{T}C = C(AB)^T = I_{n}##. The question does not say that A and B are invertible, however given that the inverse of (AB)T exists you can show that they are indeed invertible.

Sorry I left out the fact that A and B are invertible.

HallsofIvy
You have already said that $(AB)^T= B^TA^T$. Okay, what will get rid of that $B^T$ and then what will get rid of that $A^T$
(Of course, you know that a matrix is invertible if and only if its determinant is not 0? And that the determinant of $X^T$ is the same as the determinant of $X$?)