# Math Help - Prove that matrix is invertible

1. ## Prove that matrix is invertible

Hi.
I need to prove that if $A^2+5A+6I=0$ then A is invertibale.
so im not sure if thats valid, but i think i can write it like that:
$(A+2I)(A+3I)=0$

but it still doesn't prove that $A+2I=0$ or $A+3I=0$, since they're bote can be zero divisors and therefore don't necessarily equal 0.

so im kinda lost here.
is there another way?

TIA!

2. ## Re: Prove that matrix is invertible

from A2 + 5A + 6I = 0 we have:

6I = -A2 - 5A

I = (1/6)(-A - 5I)A

so evidently, A-1 = (-1/6)(A + 5I)

(not only is A invertible, we actually found the inverse!)