# Inverse matrix problem

#### dvdy

I'm having trouble with this question

A^2 + 3A - 6I = 0 where A is a square matrix.
Explain why A inverse exists and find it in terms of A.

EDIT: Looking around it seems this thread would have been more suitable for the Linear algebra section. Could someone move it? Thanks.

#### tonio

I'm having trouble with this question

A^2 + 3A - 6I = 0 where A is a square matrix.
Explain why A inverse exists and find it in terms of A.

EDIT: Looking around it seems this thread would have been more suitable for the Linear algebra section. Could someone move it? Thanks.

A matrix is singular (= non-invertible) iff zero is one of its eigenvalues iff its characteristic polynomial has free coefficient equal to zero.

Since $$\displaystyle x^2+3x-6$$ is a polynomial which vanishes at $$\displaystyle A$$ we then know that the minimal pol. of $$\displaystyle A$$ divides it, and since the min. pol. and the char. pol. of $$\displaystyle A$$ have both exactly the same irreducible factors then...

Tonio