Chapter 7.3, Problem 54E

(a)

To determine

To write: B in block form, $B=\left[\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\right],$ where $B_{11}$ is an $m\times m$ matrix.

(b)

To determine

To explain: $f_A\left(\lambda \right)=f_B\left(\lambda \right)=f_{B_{22}}\left(\lambda \right)f_{B_{11}}\left(\lambda \right)={\left(-1\right)}^m{f}_{B_{22}}\left(\lambda \right)\left({\lambda }^m-a_{m-1}{\lambda }^{m-1}-\mathrm{...}-a_1\lambda -a_0\right)$ .

(c)

To determine

To explain: $f_A\left(A\right)\overrightarrow{v}=\overrightarrow{0}$ .

(d)

To determine

To explain: $f_A\left(A\right)=0$ .