1. Consider the following Hermitian 2×2 matrices $Q$, find an invertible matrix $A$ and diagonal matrices $D$, such that $Q = A^* D A$.

• $ Q = \begin{bmatrix} 1 & 2 \cr 2 & 2 \end{bmatrix} $
• $ Q = \begin{bmatrix} 0 & i \cr -i & 0 \end{bmatrix} $
• $ Q = \begin{bmatrix} 0 & i \cr -i & 2 \end{bmatrix} $

2. Find the eigenvalues of the following matrix, and for each eigenvalue find an eigenvector. $$ T = \begin{bmatrix} 1 & 2 & 5 \cr 0 & 2 & 3 \cr 0 & 0 & -1 \end{bmatrix} $$

3. Let $A$ be the following $3 x 3$ matrix, Use Gauss Elimination to find $\det A$ and $A^{-1}$ $$ A = \begin{bmatrix} 0 & 2 & -1 \cr 1 & 2 & 1 \cr 0 & 3 & -1 \end{bmatrix} $$

4.