###### Exercise 7

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

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

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

Every vector is an eigenvector of the identity matrix.

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

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