Exercise 1

For each of the following matrices, find its characteristic polynomial, its eigenvalues, and the multiplicity of each eigenvalue.

1. $A=\left[\begin{array}{rr} 4 & -1 \\ 4 & 0 \\ \end{array}\right]$.

2. $A=\left[\begin{array}{rrr} 3 & -1 & 3 \\ 0 & 4 & 0 \\ 0 & 0 & -6 \end{array}\right]$.

3. $A = \left[\begin{array}{rr} -2 & 0 \\ 0 & -2 \\ \end{array}\right]$.

4. $A=\left[\begin{array}{rr} -1 & 2 \\ 2 & 2 \\ \end{array}\right]$.