Inverse matrix problem

• May 16th 2010, 09:59 PM
dvdy
Inverse matrix problem
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.
• May 17th 2010, 01:20 AM
tonio
Quote:

Originally Posted by 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.

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

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

Tonio