How show that if Matrix A is not square, it cannot have an inverse. How to show that if How A is not square, it cannot have an inverse. 
Why is the the case and how can I prove it? 
 A: Hint: If a matrix has an inverse, then $$AA^{-1}=A^{-1}A=I.$$
A: Non-square matrices $m \times n$ matrices for which $m ≠ n$ do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. 
Let $A$ is $m \times n$ and the rank of $A$ is equal to $n$, then $A$ has a left inverse: an $n \times m$ matrix $B$ such that $BA = I$. 
while, If $A$ has rank $m$, then it has a right inverse: an $n \times m$ matrix $B$ such that $AB = I$.
A: Say that $A$ is an $n\times m$ matrix. If $B$ is another matrix such that $AB$ and $BA$ moth make sense, then $B$ must be $m \times n$. Then $AB$ is $n \times n$ and $BA$ is $m \times m$. 
As usual, the matrix $A$ represents a linear map $f_A$ from $\mathbb{R}^m$ to $\mathbb{R}^n$ and the matrix $B$ represents a linear map $f_B$ from $\mathbb{R}^n$ to $\mathbb{R}^m$. 
If $AB$ is the identity matrix, then $f_A \circ f_B$ is the identity function, so in particular $f_A$ is surjective. That can only happen if $m \geq n$, because there is never a surjective linear map from a vector space to a different space of larger dimension.
Similarly, if $BA$ is also the identity, then $f_B$ is surjective, and thus $n \geq m$.
Putting these together, if $AB$ and $BA$ are both identity matrices then $n = m$, which means the matrices must be square.
