91Ó°ÊÓ

Prove that \(\mathrm{n} \times \mathrm{n}\) matrix \(\mathrm{A}\) is diagonalizable if and only if it has n linearly independent eigenvectors. In this case \(\mathrm{A}\) is similar to a matrix D whose diagonal elements are the eigenvalues of \(\mathrm{A}\).

Find a nonsingular matrix \(\mathrm{P}\) such that \(\mathrm{P}^{-1} \mathrm{AP}\) is diagonal, given that $$ \mathrm{A}=\begin{array}{rr} 1 & 1 \\ & \mid 3 & -1 \mid \end{array} $$