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\lhead{Mathematics 262}
\rhead{J.~Palmieri}

\begin{center}
{\large \textbf{Exam 1}} \\
\textbf{Friday the 13th (of February)}
\end{center}

General instructions: You may not use calculators.  Unless otherwise
specified, treat all matrices (etc.) as being defined over the real
numbers $\mathbf{R}$.  You may use any result from class or from the
homework, if you want to.  If I ask a question about $n\times n$
matrices and you're not sure how to do it, you will get partial credit
if you just work on the $2 \times 2$ case or the $3 \times 3$ case.
(You might even get a tiny bit of credit for the $1 \times 1$ case.)

\begin{questions}

\uplevel{\textbf{Short answer}}

\question[5]
Give a formula for the \emph{determinant} of an $n \times n$ matrix.

\question[5]
Suppose that $T$ is a linear operator on an $n$-dimensional vector
space.  Give the definition of an \emph{eigenvalue} of $T$.

\question[5]
Describe \emph{Cramer's rule}.

\uplevel{\textbf{True or false:} For the next two problems, tell me
whether each statement is true or false.  If it's true, give a
\emph{brief} reason why.  If it's false, give a counterexample.}

\question[10]
If $A$ is an $n\times n$ matrix with rank less than $n$, then $0$ is
an eigenvalue of $A$.

\question[10]
For $n\times n$ matrices $A$ and $B$, $\det (A + B) = \det (A) +
\det (B)$.

\uplevel{\textbf{Computations}}

\question
\begin{parts}
\part[10]
Compute the determinant of
$ \displaystyle 
\begin{bmatrix}
1 & -2 & \ 3 & -4 \\
5 & \ 6 & -7 & -8 \\
0 & \ 0 & \ 1 & \ 1 \\
0 & \ 0 & \ 3 & \ 4
\end{bmatrix}$.

\part[10]
Compute the inverse of
$\displaystyle 
\begin{bmatrix}
\ 3 & \ 0 & 0 \\ 
-5 & -2 & 0 \\ 
\ 4 & \ 0 & 1
\end{bmatrix}$.

\end{parts}

\question
Find the eigenvalues of the following matrices, and for each eigenvalue,
find a basis for the corresponding eigenspace.  Is each matrix
diagonalizable?  (Give a brief reason.)
\begin{parts}
\part[10]
$\displaystyle 
\begin{bmatrix}
1 & 1 & 1 \\
0 & 1 & 3 \\
0 & 0 & 2
\end{bmatrix}$

\part[10]
$\displaystyle 
\begin{bmatrix}
\ \cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{bmatrix}$ (Your answer may depend on $\theta$.)
\end{parts}

\newpage
\uplevel{\textbf{Theory}}

\question
\begin{parts}
\part[15]
Suppose that $A$ is a $2 \times 2$ matrix of real numbers which is
symmetric (i.e., $A=A^{t}$).  Prove that $A$ is similar to a diagonal
matrix.

\part
\textbf{(extra credit)} Can you extend this result to $3 \times 3$
symmetric matrices, or in general to $n \times n$ symmetric matrices?
\end{parts}

\question
\begin{parts}
\part[10]
Let $A$ be an $n \times n$ matrix with characteristic polynomial 
\[
f(x)
= x^{n} + a_{n-1}x^{n-1} + \dots + a_{1}x + a_{0}.
\]
What is the constant term $a_{0}$, in terms of $A$?  (Hint: plug $x=0$
into the characteristic polynomial.)

\part
\textbf{(extra credit)} Can you express the coefficient $a_{n-1}$ of
$x^{n-1}$ in a similar way?  What about the other coefficients?

\end{parts}

\end{questions}

\end{document}
