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\begin{center}
{\Large\bf Math 235/235J Final Examination\\
Solutions and Marking 
Scheme}\\[4mm]

\vspace{1mm}

Wednesday, December 13, 2000 \\
2:00 -- 5:00 p.m. \\ [2mm]
{\large University of Waterloo} \\ [4mm]
\end{center}

~~\\
~~\\
~~\\
~~\\
~~\\



\noindent
\rmp{[Marks]}
\begin{enumerate}

\item[1.]

\begin{itemize}

\item[(a)] If \rmp{[12]} $A$ is a $2\times 2$ matrix and Tr$(A)$ its trace,
verify that
$$
(\mbox{Tr}(A))^2-\mbox{Tr}(A^2)=2\det(A).
$$

\vspace{4.5in}

\end{itemize}

Compute the following determinants in (b) to (e):

\begin{itemize}

\item[(b)] $\det\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 2 & 6 & 0 & 0 \\ -1
& 0 & -2 & 0 \\ 0 & 2 & 0 & 5 \end{array}\right].$

\vspace{2in}

\item[(c)] $\det\left[\begin{array}{rrrr} -1 & 0 & 0 & 0 \\ 2 & 0 & 0 & -6 \\
-1 & 1 & -2 & 19 \\ 0 & 2 & 2 & 5\end{array}\right].$


\item[(d)] $\det\left[\begin{array}{rrrr} -1 & 3 & -3 & 3 \\ 2 & -6 & 6 & -6
\\ -1 & 0 & -2 & 9 \\ 0 & 2 & 12 & 5 \end{array}\right].$

\vspace{4in}

\item[(e)] $\det\left[\begin{array}{rrrr} 1 & 2 & 3 & 0 \\ 2 & 6 & 6 & 1 \\ -1
& 0 & 0 & 3 \\ 0 & 2 & 0 & 5 \end{array}\right].$ Compute this determinant by
using row reduction to obtain an upper triangular matrix.

\end{itemize}


\item[2.] Assume \rmp{[8]} that the matrices of the linear systems in (a) and
(b) below are nonsingular. Use Cramer's Rule to solve these systems for $y$
(only).

\begin{itemize}

\item[(a)] \hspace{2.5in}$ax+by=1$\\
$~~$\hspace{2.4in} $cx+dy=0$



\solution
Value is 2:
\[y=
  \frac{\det \pmatrix{a & 1\cr c & 0}}{\det \pmatrix{a & b\cr c & d}}
    =\frac{-c}{ad-bc}
\]
-1 for wrong sign (in both parts a and b).
\epr


\item[(b)] \hspace{2.25in} $ax+by+cz = 1$ \\
$~~$\hspace{2.15in} $dx+ey+fz = 0$ \\
$~~$\hspace{2.17in} $gx+hy+iz = 0$ \\[3mm]
(Denote the determinant of the matrix of this system by $D.$ You need not
expand it.)

\solution
Value is 3:
\[y=\frac{\det \pmatrix{a & 1 & c \cr d &0 &f \cr g &0 &i }}{D}
   =\frac{fg-di}{D}
\]
-1 for wrong sign (in both parts a and b).


\item[(c)] Find the value of the parameter $t$ which makes the vectors
\[[5,-1,1],[-2,1,-t],[9,3,2t]
\]
 linearly dependent.

\solution
Value is 3:\\
There are many ways to solve this, e.g. set the determinant to 0 or use
Gaussian elimination.
Marks were given if clear details were given for the equivalence of
linear dependence. Arithmetic errors were penalized but judgement was
used on how clear the rest of the solution was, e.g. no marks were given
for just stating the equivalence in terms of linear dependence and not
getting started on the arithmetic.

Set
\begin{eqnarray*}
0  &=& \det \pmatrix{5 & -2 & 9 \cr -1 &1 &3 \cr 1 &-t &2t }\\
  &=& 5(2t+3t) + (-4t+9t)+(-6-9)\\
  &=&30t-15
\end{eqnarray*}
or $t=15/30=1/2.$


\end{itemize}


\item[3.] Let \rmp{[10]} $A=\pmatrix{ 0 & 3 & 0 \cr 1 & 0 & 2 \cr 0 & 1 &
2}.$

\begin{itemize}

\item[(a)] Calculate the characteristic polynomial of $A,$ and determine the
eigenvalues of $A.$

\vspace{3.5in}

\item[(b)] Let $\lambda$ be the eigenvalue of $A$ with smallest absolute
value. Determine all eigenvectors of $A$ associated with $\lambda.$

\vspace{4.25in}

\item[(c)] Is $A$ diagonalizable? (Justify your claim.)

\end{itemize}


\item[4.]

\begin{itemize}

\item[(a)] Let \rmp{[12]} $A$ be a square matrix with eigenvalue $\lambda.$ By
using induction on $k,$ show that $\lambda^k$ is an eigenvalue of $A^k$ for
all integers $k\geq 1.$

\vspace{5.5in}

\item[(b)] Assume that $A$ satisfies the matrix equation $A^2+A-2I=0.$ Show
that if $\lambda$ is an eigenvalue of $A$ then either $\lambda=1$ or
$\lambda=-2.$


\item[(c)] Assume further that $A$ is $n\times n$ and that Tr$(A)=0.$ Show
that $n$ is divisible by 3. (Hint: Use the result from part (b) and consider
the (algebraic) multiplicities of the eigenvalues 1 and $-2$.)

\end{itemize}


\item[5.] Let \rmp{[12]} $A=\pmatrix{ 2 & 1 & 1 \cr 1 & 2 & 1 \cr 1 & 1 &
2}.$ Find an orthogonal matrix $P$ and a diagonal matrix $D$ such
that $P^{-1}AP=D.$


\item[6.] Answer \rmp{[12]} one part only.

\begin{itemize}

\item[{\bf Part I.}] Let $Q:\mathbf{R}^2\rightarrow\mathbf{R}$ be the
quadratic form defined by $Q(x,y)=-3x^2+8xy+3y^2.$

\begin{itemize}

\item[(a)] Find the symmetric matrix $A$ such that
$Q(x,y)=[x,y]A\pmatrix{ x \cr y}.$

\vspace{1.25in}

\item[(b)] Find an orthogonal matrix $P$ such that $P^{-1}AP$ is diagonal.

\vspace{5in}

\item[(c)] Sketch in the $x$--$y$ plane the conic defined by the equation
$Q(x,y)=1.$

\end{itemize}


\item[{\bf Part II.}] Let $A=\left[\begin{array}{rrr} 2 & 0 & 0 \\ 0 & -3 & 4
\\ 0 & 4 & 3 \end{array}\right].$ Determine the most general solution $x(t)$
to the system of ordinary differential equations $\dss\frac{dx}{dt}=Ax.$ Show
clearly how the solution is obtained.

\end{itemize}


\item[7.] Let \rmp{[12]} $\mathcal{M}$ be the space of 3 by 3 real matrices
and $L:\mathcal{M}\rightarrow\mathcal{M}$ the linear operator defined by
$L(X)=AX-XA$ where
$$
A=\pmatrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & 0 & 0
}.
$$
Let $T:\mathcal{M}\rightarrow\mathbf{R}^3$ be the linear transformation which
maps an arbitrary matrix $X=[x_{ij}]\in\mathcal{M}$ to
$(x_{11}+x_{22}+x_{33},x_{21}+x_{32},x_{31}).$

\begin{itemize}

\item[(a)] Compute $L(X)$ for an arbitrary $X\in\mathcal{M}$ and find a basis
for the kernel of $L.$


\item[(b)] Show that range$(L)=\mbox{kernel}(T).$

\end{itemize}


\item[8.] Let \rmp{[12]} $(\cdot,\cdot)$ be the standard inner product over
$\mathbf{C}^n.$ Let $A$ be a Hermitian matrix of size $n.$

\begin{itemize}

\item[(a)] Show that $(A\mathbf{v},\mathbf{w})=(\mathbf{v},A\mathbf{w})$ for
all $\mathbf{v},\mathbf{w}\in\mathbf{C}^n.$

\vspace{5.5in}

\item[(b)] Show that every eigenvalue $\lambda$ of $A$ is real.


\item[(c)] Show that if $\mathbf{v}$ and $\mathbf{w}$ are eigenvectors of $A$
corresponding to distinct eigenvalues $\lambda$ and $\mu$ respectively, then
$\mathbf{v}$ and $\mathbf{w}$ are orthogonal.

\end{itemize}


\item[9.] Determine \rmp{[10]} if the statements below are {\bf true} or {\bf
false}. Indicate your answers by entering {\bf T} or {\bf F} in the boxes to
the right. You do not need to justify your answers.

Each correct answer is worth 2 marks. Each incorrect answer is worth $-1$
mark. If you do not give an answer, you get 0 marks for that statement. If the
total score for this question is negative, it will be replaced by zero.

\begin{itemize}

\item[(a)] If $A$ and $B$ are $n\times n$ real matrices, then
$\det(AB)=\det(A^TB^T).$

\vspace{.5in}

\item[(b)] Similar matrices have the same eigenvectors.

\vspace{.5in}

\item[(c)] All eigenvalues of a square matrix are nonzero.

\vspace{.5in}

\item[(d)] If $A$ is a unitary matrix then $|\det(A)|=1.$

\vspace{.5in}

\item[(e)] If $\mathbf{v}$ and $\mathbf{w}$ are linearly dependent unit
vectors in a\\
complex inner product space, then $\mathbf{w}=\mathbf{v}$ or
$\mathbf{w}=-\mathbf{v}.$

\end{itemize}

\begin{flushright}

\vspace{-3.85in}

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\fbox{\rule[-5mm]{0cm}{.4cm}\hspace{1.5cm}} \\ [.5in]

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\fbox{\rule[-5mm]{0cm}{.4cm}\hspace{1.5cm}} \\ [.5in]

\fbox{\rule[-5mm]{0cm}{.4cm}\hspace{1.5cm}} \\

\end{flushright}

\end{enumerate}

\end{document}
