If $A$ and $B$ are singular and nonsingular respectively, where both are square, is $A+B$ always nonsingular?
Suppose that $A$ is a singular matrix and that $B$ is nonsingular, where both are square of the same dimension. It is not hard to see that $AB$ and $BA$ are both singular. It seems natural to ask whether the same is true for addition of matrices instead of product.
For $1\times1$ matrices (i.e., numbers), the only singular matrix is $0$; so if we add it to any nonsingular (invertible) matrix, it remains nonsingular. So to find a counterexample, we have to look at bigger matrices.
Best Answer
Not true even for positive matrices: $$ \begin{pmatrix}1 & 1\\2 & 2\end{pmatrix}+ \begin{pmatrix}3 & 2\\2 & 1\end{pmatrix}= \begin{pmatrix}4 & 3\\4 & 3\end{pmatrix}. $$