# If a matrix is row equivalent to some invertible matrix then it is invertible

I'm not really sure where to go with this question, any help would be appreciated.

## Question

Let $A$ be an $n\times n$ matrix. Prove that if $A$ is row equivalent to some invertible $n\times n$ matrix $B$ then A is invertible.

## Attempt

I'm not sure where a starting point would be. I know that an $n\times n$ matrix B is invertible if there is a matrix A such that B is both the left and the right inverse of A: AB = In and BA = In, but I'm not sure if this would be useful.

• @hjpotter92 Thanks for editing the body of the post. It's best to also edit the title along with the body, when needed (as it was here). – user147263 Nov 2 '15 at 12:52

If $A$ is row-equivalent to say $B$ and $B$ is invertible, then thre exist elementary matrices $E_1,\dots,E_r$ such that $B=E_r\dots E_1A$. Now, each $E_i$ is invertible. So, is a product of invertible matrices.