###### Exercise 4

Provide a justification for your response to the following questions.

If \(A\) is a \(3\times3\) matrix having eigenvalues \(\lambda = 2, 3, -4\text{,}\) can you guarantee that \(A\) is diagonalizable?

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

If \(A\) is similar to the matrix \(B = \left[\begin{array}{rrr} -5 \amp 0 \amp 0 \\ 0 \amp -5 \amp 0 \\ 0 \amp 0 \amp 3 \\ \end{array}\right] \text{,}\) is \(A\) diagonalizable?

What matrices are similar to the identity matrix?

If \(A\) is a diagonalizable \(2\times2\) matrix with a single eigenvalue \(\lambda = 4\text{,}\) what is \(A\text{?}\)