How show that if Matrix A is not square, it cannot have an inverse.

Question

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?

No I do not know. – Rollo Montgomery Konig-Brock Oct 05 '13 at 16:05 — Rollo Montgomery Konig-Brock, Oct 05 '13 at 16:05

score 2 · Answer 1 · answered Oct 05 '13 at 16:02

2

Hint: If a matrix has an inverse, then $$AA^{-1}=A^{-1}A=I.$$

answered Oct 05 '13 at 16:02

Clayton

24,751

I know that formula, but how to I algebraically represent a non-square matrix and use the formula? – Rollo Montgomery Konig-Brock Oct 05 '13 at 16:07
Doesn't this presuppose that $A$ and $A^{-1}$ must be square, in order for $AA^{-1}$ and $A^{-1}A$ to be equal to the same matrix $I$? – Carl Mummert Oct 05 '13 at 16:17
3

@Cameron Williams: in the non-square case, one would only expect that $AB$ and $BA$ are both identity matrices (of different sizes). The question is, why is that impossible? If we want to define that $AB$ and $BA$ have to be the same size, then the answer is just "it is a matter of definition" and there is nothing to prove. – Carl Mummert Oct 05 '13 at 16:24

score 2 · Accepted Answer · answered Oct 05 '13 at 16:18

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 $AB = I$.

score 1 · Answer 3 · answered Oct 05 '13 at 16:17

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.

How show that if Matrix A is not square, it cannot have an inverse.

3 Answers3