Showing a matrix is invertible

[noelo2014]

Homework Statement

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

Homework Equations

The Attempt at a Solution

(A.B)^T=B^T.A^T

 

[CAF123]

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.

 

[noelo2014]

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 $X$?)