# Is the sum of two invertible matrices necessarily invertible?

May 20, 2018

No

#### Explanation:

A minimal example with $1 \times 1$ matrices would be:

$\left(1\right) + \left(- 1\right) = \left(0\right)$

A less trivial example with $2 \times 2$ matrices would be:

$\left(\begin{matrix}1 & 1 \\ 0 & 1\end{matrix}\right) + \left(\begin{matrix}- 1 & 1 \\ 0 & - 1\end{matrix}\right) = \left(\begin{matrix}0 & 2 \\ 0 & 0\end{matrix}\right)$