TITLE: If $A$ is singular, is $A^3+A^2+A$ singular?
QUESTION [10 upvotes]: Suppose that $A$ is singular, is $A^3 + A^2 + A$ singular as well?

REPLY [4 votes]: A square matrix is singular if and only if there exists a nonzero vector that maps to zero when transformed by the matrix.
Let $A\in\mathbb{R}^\mathit{n{\times}n}$ be singular. Then there exists $x\in\mathbb{R}^n$ such that $x\neq0$ and $Ax=0$. Thus $(A^3+A^2+A)x=(A^2+A+I)Ax=(A^2+A+I)0=0$. Therefore  $A^3+A^2+A$ is singular.