Exercise 3

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

  1. The eigenvalues of a matrix \(A\) are the entries on the diagonal of \(A\text{.}\)

  2. If \(\lambda\) is an eigenvalue of multiplicity \(1\text{,}\) then \(E_\lambda\) is one-dimensional.

  3. If a matrix \(A\) is invertible, then \(\lambda=0\) cannot be an eigenvalue.

  4. If \(A\) is a \(13\times 13\) matrix, the charasteristic polynomial has degree less than \(13\text{.}\)

  5. The eigenspace \(E_\lambda\) of \(A\) is the same as the null space \(\nul(A-\lambda I)\text{.}\)