031 Review Part 1, Problem 5

True or false: If  $A$  and  $B$  are invertible  $n\times n$  matrices, then so is  $A+B.$

Solution:
Let  $A={\begin{bmatrix}1&1\\0&1\end{bmatrix}}$  and  $B={\begin{bmatrix}-1&-1\\0&-1\end{bmatrix}}.$
Notice that
${\text{det}}A={\text{det}}B=1.$
Since  ${\text{det}}A,{\text{det}}B\neq 0,$  we know  $A$  and  $B$  are invertible.
But,
$A+B={\begin{bmatrix}0&0\\0&0\end{bmatrix}},$
which is not invertible.
Hence, this statement is false.

Final Answer:
FALSE

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