Show that $A$ is invertible. Let $A$ be an $n \times n$ matrix that satisfies $A^3 +a_{2}A^2 +a_{1}A+I_{n} =0$

Question

Let $A$ be an $n \times n$ matrix that satisfies $A^3+a_{2}A^{2}+a_{1}A+I_{n} = 0$ where $A^{2} = AA$ and $A^{3} = AA^{2}$. Show that $A$ is invertible.

$\textbf{Hint:}$ Let $B = (A^{2} + a_{2}A+a_{1}I_{n})$ and verify that $AB=BA=I_{n}$.

I don't know how to even start this question. Please help!

Answer 1

From the given equation you get that $$I_n=A(-A^2-a_2A-a_1)=(-A^2-a_2A-a_1)A$$

This shows that the matrix in the hint (well, actually its opposite) is the inverse of $A$.