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\prof{Peter Cholak}
\class{Math 221}
\quizdate{Monday, September 16}

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\heading

\noindent For the entire quiz, we will consider the matrix
$A = 
        \begin{bmatrix}
                1 & 0 & 3 \\
                0 & 3 & 0 \\
                0 & 3 & 1 \\
        \end{bmatrix}.$
        It is to your advantage to read all of the questions
        first before beginning work \bigskip
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\begin{problem}
  Is $A$ invertible? If so, find the inverse.
\end{problem}
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\vfill
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\begin{problem}
  Solve the equation $A\mathbf{x} = \mathbf{b}$ for $\mathbf{x}$,
  where $\mathbf{x} = \begin{bmatrix}
    x \\ y \\ z \\
                                \end{bmatrix} $
and  $\mathbf{b} =
        \begin{bmatrix}
                2   \\
                -5 \\
                0  \\
        \end{bmatrix}.$
\end{problem}
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\vfill

\noindent  {\em Turn over; more problems on the reverse!}
\newpage
\noindent
Form the new matrix 
$C =    \begin{bmatrix} A & \left | 
                \begin{matrix} 
                       2  \\ -5  \\ 0  
                \end{matrix}  \right .
        \end{bmatrix} =  
        \begin{bmatrix} 
                \begin{matrix}
                  1 & 0 & 3 \\
                  0 & 3 & 0  \\
                  0 & 3 & 1 \\
                \end{matrix} & \left | 
                \begin{matrix}
                        2  \\
                        -5  \\
                        0  \\
                \end{matrix} \right .
        \end{bmatrix} $. 
\bigskip

\begin{problem}
  Compute $A^{-1}C$.
\end{problem}
\vfill
\begin{problem}
  Solve the equation $C\mathbf{y} = \mathbf{0}$ for $\mathbf{y}$,
  where $\mathbf{y} = \begin{bmatrix} x \\ y \\ z \\ w
  \end{bmatrix}$.  Be sure to give the {\em general solution} to
  the system of equations.
\end{problem}
\vfill

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