# What is the null space of an invertible matrix?

Jul 17, 2017

$\left\{\underline{0}\right\}$

#### Explanation:

If a matrix $M$ is invertible, then the only point which it maps to $\underline{0}$ by multiplication is $\underline{0}$.

For example, if $M$ is an invertible $3 \times 3$ matrix with inverse ${M}^{- 1}$ and:

$M \left(\begin{matrix}x \\ y \\ z\end{matrix}\right) = \left(\begin{matrix}0 \\ 0 \\ 0\end{matrix}\right)$

then:

$\left(\begin{matrix}x \\ y \\ z\end{matrix}\right) = {M}^{- 1} M \left(\begin{matrix}x \\ y \\ z\end{matrix}\right) = {M}^{- 1} \left(\begin{matrix}0 \\ 0 \\ 0\end{matrix}\right) = \left(\begin{matrix}0 \\ 0 \\ 0\end{matrix}\right)$

So the null space of $M$ is the $0$-dimensional subspace containing the single point $\left(\begin{matrix}0 \\ 0 \\ 0\end{matrix}\right)$.