# Is the sum of an invertible matrix with a non-invertible matrix necessarily non-invertible?

Mar 10, 2016

No.

#### Explanation:

For example:

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

Or a less trivial example:

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