A is a matrix such that 𝐴^2+3𝐴−𝐼=0. Show that 𝐴 is invertible.

Answer 1

You are given an annihilating equation for A, so by Cayley-Hamilton theorem this is the characteristic equation also satisfied by the eigenvalues of A.

λ²+3λ-1=0

This is a quadratic equation, so we know the product of the roots is -1 and the sum of the roots is +3 (we could solve for λ with quadratic formula, but it's unnecessary here).

Each root is an eigenvalue, so the product of the eigenvalues is -1. The product of a matrix's eigenvalues is equal to the determinant of the matrix, so we know det(A)=-1. Because it has a nonzero determinant, A is invertible.