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...