\documentstyle[12pt]{article}

\newcounter{xprobno}
\newcommand{\xprob}[1]{\addtocounter{xprobno}{1}
   \par\vspace{.5in}\noindent\thexprobno . (#1 points) }
\newcommand{\norm}[1]{||#1||}
\newcommand{\complex }{\Bbb C}
\newcommand{\field }{\Bbb R}

\begin{document}

\begin{flushright}
  A. Miller\\
  M340 Exam 1\\
  February 24, 1993
\end{flushright}


{\bf There is a total of 100 points.
Put your answers on separate sheets of paper.
Keep this exam,  hand in the answers and all work sheets.
}

\xprob{6}  Let
$$A=
\left[
\begin{array}{rr}
         1   &     2         \\
         1   &     1         \\
\end{array}
\right]
\;\;
B=
\left[
\begin{array}{rrrr}
         1   &     -3   &     -1   &  2   \\
         1   &     2   &     3     &  -1  \\
\end{array}
\right]
\;\;
C=
\left[
\begin{array}{rrr}
         1   &     1        \\
         -1   &    0        \\
\end{array}
\right]
$$

(a) What is $A+C$?

(b) What is $3A$?

(c) What is $AB$?

(d) What is $B^T$?

(e) What is col$_{3}(B)$?

(f) What is entry$_{2,3}(B)$?

\xprob{4} (a) What is $I_{4\times 4}$?  (b) What is $Z_{3\times 5}$?

\xprob{8} Find a matrix  $R$ in Echelon form and an
invertible matrix $M$ such that  $MA=R$  where
$$
A=
\left[
\begin{array}{rr}
         1    &     2      \\
         0    &    -1      \\
         3    &     6       \\
         1    &     2       \\
\end{array}
\right].
$$

\xprob{6} Consider the following three equations:
$x_1+x_2+x_3+x_4=1$, $x_2+x_3+x_4=2$, and $x_4+x_5=3$.

(a) Find matrices $A$ and $B$ such that these three equations are
equivalent to the single matrix equation $AX=B$.

(b) Find two solutions of this system such that neither is
scaler multiple of the other.


\xprob{10} Find the inverse of the following matrix:
$$
\left[
\begin{array}{rrrr}
         1   &     0   &     0   &    0   \\
         1   &     1   &     1   &    0   \\
        -1   &     0   &     1   &    1   \\
         0   &     0   &     1   &    0   \\
\end{array}
\right]
$$

\xprob{4} Find a $5\times 5$ elementary matrix $E$ such that
for any $5\times n$ matrix $A$, $EA$ is the matrix obtained from
$A$ by swaping third row  and fifth row of $A$.


\xprob{10} Let
$$A=
\left[
\begin{array}{rrr}
         1   &     2   &     3      \\
         1   &     1   &     1      \\
        -1   &     0   &     1      \\
\end{array}
\right].
$$
Find a $3\times 3$ matrix B which is not the zero matrix $Z$ but such
that $AB=Z$.

\xprob{10} Let
$$A=
\left[
\begin{array}{rrr}
         0   &     1   &     1      \\
         0   &     1   &     0      \\
         1   &     1   &     0      \\
\end{array}
\right].
$$
Write $A^{-1}$ as the product of elementary matrices.

\xprob{4} Suppose $A$ and $B$ are $5\times 5$ matrices, $\det(A)=3$ and
$\det(B)=7$.  What is $\det(A^{-1}B^T)$?  If it is impossible to be
determined by this information, say so.

\xprob{4} Suppose $C$ is a $5\times 5$ matrix with $\det(C)=3$.
What is $\det(2C)$?  If it is impossible to be
determined by this information, say so.

\xprob{4} Suppose $A$ and $B$ are $5\times 5$ matrices, $\det(A)=3$ and
$\det(B)=-3$.  What is $\det(A+B)$?  If it is impossible to be
determined by this information, say so.

\xprob{6} What is the $\det(A)$ where
$$
A=
\left[
\begin{array}{rrr}
         0    &     1    &     1    \\
         1    &     0    &     1    \\
         1    &     1    &     0    \\
\end{array}
\right]
$$

\xprob{12} Let $A$ be an ${m\times n}$ matrix,  $B$ be an ${n\times p}$
matrix.

(a) State the definition of $C=AB$.

(b) Let $c$ be a scalar. Prove that $A(cB)=(cA)B$.

\xprob{12} Let $A$ and $B$ be $n\times n$ matrices such that
$$(A+B)^2=A^2+2AB+B^2.$$
Prove that $AB=BA$.

\bigskip

\noindent Hint: What is the simpliest formula you can get
for $(A+B)^2$ without the assumption that $AB=BA$?



\end{document}