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\title[Linear Algebra]
{Exam 3\\Math 3013 Section 002, Spring 2009}
 
\author[Weiping Li]{Instructor: Weiping Li}
 
\maketitle

{\bf Print Name and Student $\#$}
\vspace{4ex}
\medskip

{\bf SHOW WORK FOR CREDIT !!! SHOW WORK FOR CREDIT !!!}
\medskip

\begin{enumerate}
\item (10pts) Let $u$ and $v$ be vectors in an inner-product space, and suppose that $\|u\|=3,
\|v\| =2$. Find $\langle u+2v, u-2v\rangle$.
\vspace{1.5in}

\item (10pts) Find the area of the triangle with vertices $(2, 1, -3), (3, 0, 4)$ and $(1, 0, 5)$ in $R^3$.
\newpage

\item (10pts) Let $A$ be a $5\times 5$ matrix with $\det A = 2$.  Find the following.

(i) $\det (A + 2A)$
\vspace{0.5in}

(ii) $\det (A^T \cdot A)$
\vspace{0.5in}

(iii) $\det (A^{-2})$.
\vspace{0.2in}

\item (15pts) Let $A = \left( \begin{array}{ccc} 0&1&-1\\-1&-3&-2\\2&-2&3 \end{array}\right)$.

(i) Find the cofactor of $-3$.
\vspace{1in}

(ii) Compute $\det A$ by using pivots and row-operations {\bf ONLY}.
\newpage

\item (10pts) (i) Find the adjoint of the matrix $\left( \begin{array}{cc} -1&2\\ -3 & 4 \end{array} \right)$.
\vspace{1.5in}

(ii) Find the $x_2$ {\bf only by using Cramer's rule} for 
\[ x_1+2x_2 = 1, \ \ \ \  2x_1 +3x_2 = -2.\]
\vspace{2.5in}

\item (10pts)  Find the characteristic polynomial of $\left( \begin{array}{ccc}
1&0&0\\ -8&4&-6\\ 8&1&9  \end{array} \right)$, and eigenvalues of the matrix.
\newpage


\item (15pts) Let $A = \left( \begin{array}{ccc}
2&1&0\\-1&0&1\\1&3&1 \end{array} \right)$. The characteristic polynomial of $A$ is
given by $(\lambda +1)(\lambda -2)^2$.

(i) Find the algebraic multiplicities of $\lambda_1 = -1$ and $\lambda_2 = 2$.
\vspace{1in}

(ii) Find the geometric multiplicities of $\lambda_1 = 1$ and $\lambda_2 = -3$.
\vspace{4in}

(i) Is the matrix $A$ diagonalizable ?
\vspace{0.1in}
\newpage

\item (10pts) A matrix $A_{3\times 3}$ has an eigenvalue $\lambda_1=2$ with eigenvector $v_1=(1, 1, 0)$, eigenvalue $\lambda_2 = -1$ with eigenvector $v_2 = (0, 1, 1)$ and eigenvalue $\lambda_3 = 3$
with eigenvector $v_3=(1,1,1)$.
Compute $A^k$.
\newpage

\item (10pts) Solve the system of linear differential equations:
\begin{eqnarray*}
x_1^{'} & = &x_1-3x_2+3x_3\\
x_2^{'} & = & -5x_2+6x_3 \\
x_3^{'} & = & -3x_2+4x_3,
\end{eqnarray*}
where the matrix $\left( \begin{array}{ccc} 1&-3&3\\0&-5&6\\0&-3&4 \end{array} \right)$ has eigenvalues $1, 1, -2$.
\end{enumerate} 
\end{document}