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\begin{document}





\author{B. Hall and R. Smarandache}

\title{Math 228, Test 1}

\date{September 19, 2000}

\maketitle







\textbf{Instructions}: You have one hour for the exam. There are 11 problems,

worth 9 points each, with one free point, for a total of 100 points possible.

For multiple-choice problems, please mark your answer clearly. For all other

problems, please \textit{show your work completely}. Partial credit will be

given for all non-multiple-choice problems.



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  For each of the following matrices, state whether or not it is an

elementary matrix. If it is, state the corresponding elementary row

operation.

\begin{align*}

A  & =\left[

\begin{array}

[c]{cc}%

1 & 0\\

0 & 0

\end{array}

\right]  ;\quad B=\left[

\begin{array}

[c]{rc}%

0 & 1\\

-1 & 0

\end{array}

\right]  ;\quad C=\left[

\begin{array}

[c]{cr}%

1 & -2\\

0 & 1

\end{array}

\right]  ;\\

D  & =\left[

\begin{array}

[c]{cc}%

0 & 1\\

1 & 0

\end{array}

\right]  ;\quad E=\left[

\begin{array}

[c]{rc}%

-1 & 0\\

1 & 1

\end{array}

\right]  .

\end{align*}

\pagebreak 



\item  Let $A$ be the matrix

\[

A=\left[

\begin{array}

[c]{cccc}%

2 & 105 & 7 & 2\\

0 & 1 & 10 & 3\\

0 & 0 & 5 & 4\\

0 & 0 & 0 & 3

\end{array}

\right]  .

\]

What is the entry in the second row and fourth column of $A^{-1}?$ (Hint: use

the formula for $A^{-1}$ in terms of adj$\left(  A\right)  .$)\bigskip



a) $\frac{5}{3}\qquad$b) 0\qquad c) 50\qquad d) $-\frac{105}{16}\qquad$e)

$\frac{2}{3}$\pagebreak 



\item  Let $A$ and $B$ be the matrices

\[

A=\left[

\begin{array}

[c]{ccc}%

1 & 3 & -2

\end{array}

\right]  ;\quad B=\left[

\begin{array}

[c]{c}%

2\\

0\\

4

\end{array}

\right]  .

\]

Compute $AB$ if defined; otherwise state ``undefined.'' Do the same for $BA.$\pagebreak 



\item  What is the value of the following determinant?

\[

\det\left[

\begin{array}

[c]{rrrr}%

3 & 4 & -1 & 2\\

1 & 0 & 4 & 8\\

-1 & 0 & 1 & 1\\

2 & 0 & -1 & 1

\end{array}

\right]

\]

\bigskip



a) $-6$\qquad b) 6\qquad c) 24\qquad d) $-24$\qquad e) 0\pagebreak 



\item  What is the value of $z$ if

\[

\left[

\begin{array}

[c]{rrr}%

1 & 2 & -1\\

3 & 4 & -2\\

1 & 1 & 1

\end{array}

\right]  \left[

\begin{array}

[c]{c}%

x\\

y\\

z

\end{array}

\right]  =\left[

\begin{array}

[c]{r}%

1\\

0\\

-1

\end{array}

\right]  ?

\]

\bigskip



a) 1\qquad b) $-\frac{1}{3}\qquad$c) 3\qquad d) $\frac{1}{3}\qquad$e) $-1$\pagebreak 



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

\[

A=\left[

\begin{array}

[c]{ccc}%

2 & 4 & 2\\

1 & 3 & 0\\

2 & 2 & 2

\end{array}

\right]  .

\]

\pagebreak 



\item  Let $A$ be the matrix given in Problem 6. Find the L-U decomposition of

$A.$ No row interchanges should be necessary.\pagebreak 



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

Which of the following statements are always true?



\qquad(I) $AB$ is invertible



\qquad(II) $AB=BA$



\qquad(III) $\left(  \left(  AB\right)  ^{-1}\right)  ^{T}=\left(

A^{-1}\right)  ^{T}\left(  B^{-1}\right)  ^{T}$



\qquad(IV) $\det\left(  A+B\right)  =\det\left(  A\right)  +\det\left(

B\right)  \bigskip$



a) I only\qquad b) I, III, and IV only\qquad c) I and III only\qquad d) I, II,

and III only\qquad e) I and IV only\vspace{2.2in}



\item  The following matrix results from performing Gaussian elimination on

the augmented matrix for a system of three equations in three unknowns.

\[

\left[

\begin{array}

[c]{cccc}%

1 & 2 & 3 & 0\\

0 & 0 & 1 & 0\\

0 & 0 & 0 & 0

\end{array}

\right]

\]

How many arbitrary constants will there be in the solution of the system?\bigskip



a) One\qquad b) Two\qquad c) Three\qquad d) None, there is a unique solution\qquad



e) The system is inconsistent\pagebreak 



\item  Suppose we know that

\[

\det\left[

\begin{array}

[c]{cccc}%

a & b & c & d\\

e & f & g & h\\

i & j & k & l\\

m & n & o & p

\end{array}

\right]  =15.

\]

What is the value of the determinant

\[

\det\left[

\begin{array}

[c]{cccc}%

e & f & g & h\\

a & b & c & d\\

2i & 2j & 2k & 2l\\

-m & -n & -o & -p

\end{array}

\right]  \text{?}

\]

\bigskip



a) 30\qquad b) $-30\qquad$c) $-15\qquad$d) 15\qquad e) $\frac{15}{2}$\vspace{2.5in}



\item  The following statements are always true, except for one. Which is the

statement that is not always true? ($A$ is an $n\times n$ matrix.)\bigskip



(a) If $\det A=1$ then $A$ is invertible.



(b) If $A$ is invertible then $\det A\neq0.$



(c) If $\det A=0$ then one row of $A$ is all zeros.



(d) If one row of $A$ is all zeros then $\det A=0.$

\end{enumerate}

\end{document}