# Showing a matrix is invertible

• noelo2014
In summary, the conversation discusses how to prove that (AB)^T is invertible by finding its inverse. The method involves finding a matrix C such that (AB)^T C = C (AB)^T = I_n. It is mentioned that A and B are invertible and the fact that the determinant of a matrix is not 0 if and only if the matrix is invertible is also brought up. The conversation ends by discussing how the determinants of X^T and X are equal.
noelo2014

## 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:
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.

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 $X$?)

## What is a matrix invertible?

A matrix is considered invertible if there exists another matrix that, when multiplied, results in the identity matrix (a matrix with ones on the main diagonal and zeros everywhere else). In other words, the inverse matrix "undoes" the original matrix.

## How do I know if a matrix is invertible?

There are a few ways to determine if a matrix is invertible. One way is to calculate the determinant of the matrix. If the determinant is non-zero, then the matrix is invertible. Another way is to use the row reduction method to see if the matrix can be transformed into the identity matrix. If it can, then the matrix is invertible.

## What happens if a matrix is not invertible?

If a matrix is not invertible, it is considered singular. This means that there is no way to "undo" the matrix and obtain the identity matrix. In terms of practical applications, this can mean that the system of equations represented by the matrix has no unique solution.

## Can a matrix be partially invertible?

No, a matrix is either invertible or not. There is no concept of partial invertibility.

## Can all types of matrices be invertible?

No, only square matrices can be invertible. This means that the number of rows is equal to the number of columns. Non-square matrices are not invertible.

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