Exercise 4

Provide a justification for your response to the following questions.

1. If $A$ is a $3\times3$ matrix having eigenvalues $\lambda = 2, 3, -4$, can you guarantee that $A$ is diagonalizable?

2. If $A$ is a $2\times 2$ matrix with a complex eigenvalue, can you guarantee that $A$ is diagonalizable?

3. If $A$ is similar to the matrix $B = \left[\begin{array}{rrr} -5 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 3 \\ \end{array}\right]$, is $A$ diagonalizable?

4. What can you say about a matrix that is similar to the identity matrix?

5. If $A$ is a diagonalizable $2\times2$ matrix with a single eigenvalue $\lambda = 4$, what is $A$?