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\begin{document}





\author{B. Hall and D. Nicholls}

\title{Math 228, Test 1}

\date{September 25, 2001}

\maketitle







\textbf{Instructions}: You have one hour for the exam. There are 10 problems

worth 10 points each for a total of 100 points possible. On all problems

except Problem 6, please \textit{show your work completely} and partial credit

will be given. For Problem 6, each item is worth two points, with no partial

credit within items.



Calculators, notes, and books are prohibited.



This exam is bound by the provisions of the Notre Dame Honor Code.\bigskip



\textbf{Name}: \bigskip



\begin{enumerate}

\item  Suppose that $E$ is an $n\times n$ elementary matrix corresponding to

the operation of interchanging two rows. What is $E^{-1}$? What is $E^{37}$? Explain.



\item  Let $A$ be the matrix

\[

A=\left[

\begin{array}

[c]{rrrr}%

0 & 3 & 1 & 18\\

0 & -9 & 1 & 8\\

0 & 0 & 0 & 2\\

2 & 22 & 3 & -14

\end{array}

\right]  .

\]

Find the entry in the second row and the first column of $A^{-1}.$

\textit{Hint}: Use the formula for $A^{-1}$ in terms of $adj(A).$



\item  Compute the value of the following determinant.

\[

\det\left[

\begin{array}

[c]{ccc}%

a & b & c\\

d & e & f\\

a+2d & b+2e & c+2f

\end{array}

\right]

\]

Explain your reasoning clearly.



\item  Find the $L$-$U$ decomposition of the matrix

\[

A=\left[

\begin{array}

[c]{rrr}%

2 & 1 & 3\\

-1 & 0 & 2\\

2 & 2 & -3

\end{array}

\right]  .

\]

No row interchanges should be needed.



\item  Suppose that $A$ and $B$ are $n\times n$ invertible matrices. Simplify

the following expression:

\[

\left(  A(BA)^{-1}B(AB)^{-1}A^{2}\right)  ^{-1}.

\]



\item  Suppose that $A$ and $B$ are invertible $n\times n$ matrices. Mark each

of the following statements as \textbf{T} (always true) or \textbf{F} (not

always true).



\_\_\_\_ $\left(  AB\right)  ^{-1}=A^{-1}B^{-1}$



\_\_\_\_ $\det\left(  \left(  AB\right)  ^{-1}\right)  =\det\left(

A^{-1}B^{-1}\right)  $



\_\_\_\_ $\det A>0$



\_\_\_\_ $\det(2A)=2\det A$



\_\_\_\_ $\det(I+A)=1+\det A$



\item  Find the general solution of the system of equations represented by the

following augmented matrix:

\[

\left[

\begin{array}

[c]{rrrr}%

2 & 1 & 4 & 0\\

2 & 2 & 5 & 1\\

2 & -1 & 2 & -2

\end{array}

\right]  .

\]



\item  Suppose $A$ is a $2\times2$ matrix and that $A^{2}=0$. (Here $0$ means

the zero matrix, all of whose entries are zero.) Is it necessarily true that

$A=0$? If so, explain why. If not, give an example of a $2\times2$ matrix $A$

such that $A^{2}=0$ but $A\neq0.$



\item  What is the value of the following determinant?

\[

\det\left[

\begin{array}

[c]{rrrr}%

1 & 2 & 57 & -9\\

2 & 1 & 8 & 43\\

0 & 0 & -1 & 1\\

0 & 0 & 4 & 1

\end{array}

\right]

\]



\item  Compute $A^{-1}$ if $A$ is the matrix

\[

A=\left[

\begin{array}

[c]{rrr}%

1 & 2 & 2\\

0 & 2 & 1\\

0 & -2 & 1

\end{array}

\right]  .

\]

\end{enumerate}

\end{document}