Let A $\in$ Mat$_n (\mathbb{F})$ and let $f(x) = a_n x^n+\cdots+a_1 x+a_0$ be the characteristic polynomial of A. Prove that A is singular if and only if $a_{0} \neq 0$.
Any hint or technique.
[Math] Singular Matrix and Characteristic Polynomial
linear algebra
Best Answer
The characteristic polynomial of $A$ is $f(x) = \det (xI-A)$. Since $\det cA = c^n \det A$ for a constant $c$, we have $f(0) = a_0 = \det (-A) = (-1)^n \det A$.
Hence $A$ is singular iff $\det A = 0$ iff $a_0 = 0$ .