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\prof{Peter Cholak}
\class{Math 221}
\quizdate{Wednesday, September 25}
\examname{Exam 1}

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\heading

\noindent  This exam is worth 100 points. You should have 7 problems.
\bigskip


\begin{problem}
  (10 points total) Let $A =
               \begin{bmatrix}
                  1 & 0 & 0 & 0 \\      
                  0 & 1 & 0 & 3 \\
                  0 & 0 & 2 & 0 \\
                  0 & 2 & 1 & 6 \\  
               \end{bmatrix}$.  

\begin{subproblem} (5 points)
   Find $\det(A)$.  Justify your answer.
\end{subproblem}
\vfill

\begin{subproblem} (5 points)
        Does $A$ have an inverse? Why?
\end{subproblem}

\newpage
\begin{problem} (30 points total) It is to your advantage to read all
of the parts of this problem before beginning. Let
  $$A =
                \begin{bmatrix}
                  0 & 0 & 0 & -1 \\
                  0 & 1 & 4  & 0 \\
                  0 & 0 & 3 & 0 \\
                  2 & 0 & 0 & 2 \\
               \end{bmatrix}.$$
  \begin{subproblem}
    (10 points) Put the matrix $A$ into reduced row echelon
    form by using {\it{only}} elementary row operations.  Show
    all your work and explicitly write out the elementary row
    operations in the order used. (This can be done using 6
    elementary row operations.) 
  \end{subproblem}

\newpage

  \begin{subproblem}
    (5 points) For each one of the above elementary row operations,
    find a matrix $E_i$ such that the matrix $E_iB$ is the matrix
    that results from doing the $i$th elementary row operation
    to the matrix $B$.
  \end{subproblem}

\vfil \vfil

  \begin{subproblem}
    (2 points) $A$ has an inverse. Why?
  \end{subproblem}

\vfil

  \begin{subproblem}
    (3 points) Write $A^{-1}$ as the product of the above elementary matrices.
  \end{subproblem}

\vfil

  \begin{subproblem}
    (3 points) Compute $\det(A)$ and $\det(A^{-1})$. (HINT:
compute $\det(A^{-1})$ first.)
  \end{subproblem}

\vfil

  \begin{subproblem}
    (3 points) Solve the system $A^{-1}\mathbf{x} = \mathbf{b}$, where
    $\mathbf{x} = 
     \begin{bmatrix}
       x_1 \\ x_2 \\ x_3 \\ x_4 \\
     \end{bmatrix}$
     and $\mathbf{b}=
       \begin{bmatrix}
         b_1 \\ b_2 \\ b_3 \\ b_4 \\
      \end{bmatrix}$
  \end{subproblem}
\end{problem}

\newpage

\vfil



\end{problem}
\vfil


\newpage



\vfil
\begin{problem}
  (10 points) Suppose that $C$ is a $3 \times 3$ matrix.  Let $E$
  be the appropriate product of elementary matrices so that $E C$
  is in reduced row echolon form. In this case, assume that
\[
E = \begin{bmatrix} 0 & 3 & -2 \\ 1 & 0 & 0 \\ 0 & 2 & 5 \\ 
\end{bmatrix} \text{ and } E C = \begin{bmatrix} 1 & 1 & 0 \\ 0 &
  0 & 1 \\ 0 & 0 & 0 \end{bmatrix}
\]
Form the new 3 by 4 matrix $A = \begin{bmatrix} C &
  \begin{matrix} 1 \\ 0 \\ 1 \end{matrix} \end{bmatrix}$, 
and solve the system of equations $A \begin{bmatrix} x_1 \\ x_2
  \\ x_3 \\ x_4 \end{bmatrix} = 0$.  Be sure to give all of the
solutions.
\end{problem}
\vfill

\begin{problem}  (15 points total)
  Consider the matrix $B =\begin{bmatrix}
    2&  0&  2&\\
    -1& 1&  0&\\
    3 & 0&  1 &\\
   \end{bmatrix}$.

\begin{subproblem} (10 points)
        Compute the characteristic polynomial $\det ( \lambda I - B)$.  
\end{subproblem}

\vfill
\begin{subproblem} (5 points)
        For what values of $\lambda$ does the equation $B \mathbf{x} = \lambda
  \mathbf{x}$ have non-trivial solutions?
\end{subproblem}
\vfil
\end{problem}

\newpage 
\begin{problem}
  (20 points total) Assume that the polynomial function $f(x)$ has the
  following form: $f(x) = ax^2 + bx + c$.

\begin{subproblem} (5 points)
   Write down the system of linear equation that $a$,
  $b$ and $c$, must satisfy in order that the graph of $f$ pass
  through the points $P= (-1,-1)$, $Q = (1,1 )$, and $R = (2,0)$.
\end{subproblem}
\vspace{2in}

\begin{subproblem} (15 points)
  Find all the functions $f$ of the given form which
  pass through the points $P$, $Q$ and $R$? 
\end{subproblem}
\end{problem}
\vfil



\newpage
\noindent Decide whether each of the following statements is True or
False, and give an explanation.
\bigskip

\begin{problem}
  (7 points) (True/False and why) If $A$ is
  an invertible matrix, and $B$ and $C$ are matrices so that
  $BA = CA$, then $B = C$.
\end{problem}
\vfill

\begin{problem}
  (8 points)(True/False and why)   The sum of two
  invertible matrices is again an invertible matrix.
\end{problem}
\vfill


\end{document}
