In Problems 31–33 A represents a 2 x 2 matrix.

Use the definitions of eigenvalue and eigenvector (Section 7.3) to prove that if  λ is an eigenvalue of A with associated eigenvector v, then -λ is an eigenvalue of the matrix -A with associated eigenvector v. Conclude that if A has positive eigenvalues 0 < λ₂ < λ₁ with associated eigenvectors v1 and v2, then -A has negative eigenvalues -λ₁ < -λ₂ < 0 with the same associated eigenvectors.