Exercise5

For each of the following matrices, use Sage to determine its eigenvalues, their multiplicities, and a basis for each eigenspace. For which matrices is it possible to construct a basis for $$\real^3$$ consisting of eigenvectors?

1. $$A = \left[\begin{array}{rrr} -4 \amp 12 \amp -6 \\ 4 \amp -5 \amp 4 \\ 11 \amp -20 \amp 13 \\ \end{array}\right]$$

2. $$A = \left[\begin{array}{rrr} 1 \amp -3 \amp 1 \\ -4 \amp 8 \amp -5 \\ -8 \amp 17 \amp -10 \\ \end{array}\right]$$

3. $$A = \left[\begin{array}{rrr} 3 \amp -8 \amp 4 \\ -2 \amp 3 \amp -2 \\ -6 \amp 12 \amp -7 \\ \end{array}\right]$$

in-context