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 $A$.

2. If $\lambda$ is an eigenvalue of multiplicity $1$, 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 characteristic polynomial has degree less than $13$.

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