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Problems #4 on pg.121 and #1, pg.167 are graded.
#4,pg.121: Compute the product of partitioned matrices $\begin{bmatrix} I & 0 \\ -E & I \end{bmatrix} \begin{bmatrix} W & X \\ Y & Z \end{bmatrix}$ (assume the partitions chosen make sense for multiplication).
Solution: Compute the same way as "normal" matrix multiplication:
$$\begin{bmatrix} I & 0 \\ -E & I \end{bmatrix} \begin{bmatrix} W & X \\ Y & Z \end{bmatrix} = \begin{bmatrix} IW+0Y & IX+0Z \\ -EW+IY & -EX+IZ \end{bmatrix}=\begin{bmatrix} W & X \\ -EW+Y & -EX+Z \end{bmatrix}.$$
#1, pg.167: Compute $\det \begin{bmatrix} 3&0&4 \\ 2&3&2 \\ 0&5&-1 \end{bmatrix}$.
Solution: Compute
$$\begin{array}{ll}
\det \begin{bmatrix} 3&0&4 \\ 2&3&2 \\ 0&5&-1 \end{bmatrix} &= 3 \det \begin{bmatrix} 3 & 2 \\ 5 & -1 \end{bmatrix} - 0 \det \begin{bmatrix} 2&2 \\ 0&-1 \end{bmatrix} + 4 \det \begin{bmatrix} 2 & 3 \\ 0 & 5 \end{bmatrix} \\
&= 3(-3-10)-0(-2-0)+4(10-0) \\
&= -39-0+40 \\
&= 1.
\end{array}$$
