Consider the upper triangular matrix
\[ A = \begin{bmatrix} 1 & 3 & 1 \\ 0 & 2 & 4e \\ 0 & 0 & 3 \end{bmatrix} \in M_3(\mathbb{R}) \]
(a) Determine the eigenvalues of A. 1, 2, 3
(b) For each eigenvalue of A, determine the corresponding eigenspace. \( E_1 = \{ t[1] : t \in \mathbb{R} \}, E_2 = \{ t[] : t \in \mathbb{R} \}, E_3 = \{ u[3] : t \in \mathbb{R} \} \)
(c) Determine real matrices B and D with B invertible and D diagonal, such that \( B^{-1}AB = D \).
\[ B = \begin{bmatrix} 1 & 3/2 \\ 0 & 1 \\ 4 & 0 \end{bmatrix} \quad \text{and} \quad D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 3 \end{bmatrix} \]
(d) Determine the trace of \( A^3 \). 36
