Inverse matrix problem

May 2010
1
0
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.
 
Oct 2009
4,261
1,836
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