Exercise 7

Determine whether the following statements are true or false and provide a justification for your response.

  1. The eigenvalues of a diagonal matrix are equal to the entries on the diagonal.

  2. If \(A\vvec=\lambda\vvec\text{,}\) then \(A^2\vvec=\lambda\vvec\) as well.

  3. Every vector is an eigenvector of the identity matrix.

  4. If \(\lambda=0\) is an eigenvalue of \(A\text{,}\) then \(A\) is invertible.

  5. For every \(n\times n\) matrix \(A\text{,}\) it is possible to find a basis of \(\real^n\) consisting of eigenvectors of \(A\text{.}\)