What is the inverse of a $1 \times 1$ matrix?

Question

What is the inverse of a $1 \times 1$ matrix?

I came across this question while studying the inverse of matrices.

If we have a $1 \times 1$ matrix, does it have an inverse? What could it be?

Answer 1

Let be $A\in\mathbb{R}^{1\times 1}$. Then for $a\in\mathbb{R}$ we can write $A=(a)$. The matrix is invertible if there exists a matrix $B\in\mathbb{R}^{1\times1}$ such that $AB=(1)$. Insteat of choosing $B$ you can choose $b\in\mathbb{R}$ and identify $B=(b)$. We get $$ (1)=AB=(a)(b)=(ab). $$ You see that it is a special and simple situation. If $a\neq 0$ you can choose $b=\frac1a$ and you got your inverse matrix.

If $(a)=A\neq (0)$ your matrix is invertable with $A^{-1}=\left(\frac1a\right)$.

Answer 2

if $a \neq 0$, $$\frac1a \times a = 1$$

Do you see the solution?