\be
\item Let $\ds A= \begin{bmatrix} 1& 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}$
and $\ds B= \begin{bmatrix} 1& 2 \\ 3 & 4 \end{bmatrix}$. Compute the product
$AB$ by partitioning both $A$ and $B$ in two different ways.

\item Suppose that $A$ is a 2-by-3 matrix, $C$ and $E$ are row vectors
in $\R^3$ and $\R^2$, respectively, $B$ and $F$ are 2 by 2 matrices. Let
$\ds M=\begin{bmatrix} A & B \\ C & E \end{bmatrix}$
and $\ds N:=\begin{bmatrix} A^t \\ F \end{bmatrix}$, where $A^t$ denotes
the transpose of $A$. Calculate $MN$ in
terms of $A$, $B$, $C$, $E$, and $F$.

\item Suppose that $a$ and $b$ are scalars, $\vecx$ and $\vecy$ are row
vectors in $\R^3$, $\vecu$ and $\vecv$ are (column) vectors in $\R^3$, and $A$ and $B$
are 3-by-3 matrices. Let
$\ds M=\begin{bmatrix} a & \vecx \\ \vecv & A \end{bmatrix}$
and $\ds N=\begin{bmatrix} b & \vecy \\ \vecu & B \end{bmatrix}$. Calculate $MN$ in
terms of $a$, $b$, $\vecx$, $\vecy$, $\vecu$, $\vecv$, $A$, and $B$.

\item Suppose that $A$ and $B$ are $n$ by $n$ matrices and that $0_n$ denotes the $n$ by $n$ zero matrix. Show that $A$ and $B$ are invertible if and only
    if $\ds M=\begin{bmatrix} A & 0_n \\ 0_n & B \end{bmatrix}$ is invertible.
\item For each of parts (a) and (c) of Exercise 1 in Section \ref{S:Simplex},
calculate the vector, $\vecy_1$, and matrix, $M_1$, corresponding to the first iteration of the simplex algorithm.

\item For the \LP LP, calculate the vector, $\vecy_3$, and matrix, $M_3$, corresponding to the third
and final iteration of the simplex algorithm. Then, calculate the final tableau matrix, using the vectors and matrices,
$M_1$, $M_2$, $M_3$, $\vecy_1$, $\vecy_2$, and $\vecy_3$.

\ee

