Answer
True.
Work Step by Step
The 2x2 matrix $A$ is invertible if and only if $ad-bc\neq 0$.
If $ad-bc=0$, then $A$ does not have a multiplicative inverse.
If $A=\left[\begin{array}{ll}
a & b\\
c & d
\end{array}\right]$, then $A^{-1}=\displaystyle \frac{1}{ad-bc}\left[\begin{array}{ll}
d & -b\\
-c & a
\end{array}\right]$.
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So, for example,
$\left[\begin{array}{ll}
1 & 0\\
0 & 1
\end{array}\right]$ and $\left[\begin{array}{ll}
-1 & 0\\
0 & -1
\end{array}\right]$ both have inverses,
but their sum, $\left[\begin{array}{ll}
0 & 0\\
0 & 0
\end{array}\right]$ does not.
True.