Problem C20:Find the eigenvalues, eigenspaces, algebraic multiplicities and geometric multiplici-ties for the matrix below. It is possible to do all these computations by hand, and it would be instructive to do so.

\[ B= \begin{pmatrix} -12 & 30 \\ -5 & 13\\ \end{pmatrix} \] \[ pB(\lambda) = det(B-(\lambda I2)) \] \[ = \begin{pmatrix} -12-\lambda & 30 \\ -5 & 13-\lambda \\ \end{pmatrix} \] \[ = (-12-\lambda)(13-\lambda)-(30)(-5) \] \[ = -156+12\lambda-13\lambda+\lambda ^2 \ +150 \]

\[ = \lambda ^2-\lambda-6 \]

\[ =(\lambda-3)(\lambda+2) \]

Eigenvalues

\[\lambda = 3 and \lambda = -2 \]

Algebraic multiplicities

\[ \alpha _B(3) = 1 \]

\[ \alpha _B(-2) = 1 \]

Eigenspace

if \[\lambda=3\] then B-3I2 =

\[ = \begin{pmatrix} -15 & 30\\ -5 & 10 \end{pmatrix} \]

\[ RREF = \begin{pmatrix} 1 & -2\\ 0 & 0 \end{pmatrix} \] \[ = \Bigg\langle \Bigg\{ \Bigg[ \begin{pmatrix} &2\\ &-1\\ \end{pmatrix} \Bigg] \Bigg\} \Bigg\rangle \]

if
\[\lambda=-2\] then B+2I2 =

\[ = \begin{pmatrix} -10 & 30\\ -5 & 15 \end{pmatrix} \] \[ RREF = \begin{pmatrix} 1 & -3\\ 0 & 0 \end{pmatrix} \]

\[ = \Bigg\langle \Bigg\{ \Bigg[ \begin{pmatrix} &3\\ &1\\ \end{pmatrix} \Bigg] \Bigg\} \Bigg\rangle \]

Geometric multiplicities \[\gamma_B(3)=1\] and \[\gamma_B(-2)=1\]