# Matrix inverses.
Let $A$ be an $n\times k$ matrix. We say $X$ is a left inverse of $A$ if $XA$ is an identity matrix, and we say $Y$ is a right-inverse of $A$ if $AY$ is an identity matrix.
Since identity matrices are square, we see that if $X$ is a left-inverse of $A$, then $XA = I_{k}$ and $X$ is $k\times n$. And if $Y$ is a right-inverse $A$, then $AY=I_{n}$ and $Y$ is $k\times n$.
We claim, if $A$ has a left-inverse $X$ and a right-inverse $Y$, then $X = Y$.
Indeed, $X = X I_{n} = X A Y =(XA)Y = I_{k}Y=Y$.
$$\def\mat#1{\begin{bmatrix}#1\end{bmatrix}}
\def\augmat#1#2{
\left[
\begin{array}{c | c}
\begin{array}{} #1
\end{array} &
\begin{array}{}
#2
\end{array}
\end{array}
\right]}$$
$\augmat{1 & 1 &3\\ 2 & 7 & 4}{3\\2}$