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\begin{center}
{\Large TEST 3

Math 315}
\end{center}

\begin{itemize}
\item  Show all your work and make your work neat.  You will be graded on this!
\item  You will be graded on both your answer and the work done to get that answer.
\item  You can use your calculator, but I should be able to follow your work as if you had no calculator!
\item  Do your work on other paper.
\item  Each problems is worth 10 points each.
\end{itemize}

For the entire test, consider the matrices below:
\[
A=\left[
  \begin{array}{rrr}
    2 & 4 & 1 \\
    0 & 0 & 1 \\
    1 & 2 & 0
  \end{array}
  \right]
\qquad
B=\left[
  \begin{array}{rrr}
    2 & 4 & 1 \\
    1 & 0 & 1 \\
    1 & 2 & 5
  \end{array}
  \right]
\qquad
C=\left[
  \begin{array}{rr}
    0 & 4 \\
    -1 & -4 
  \end{array}
  \right]
\]




\begin{enumerate}

\item  
  \begin{enumerate}
  \item  What is the determinant of $A$?  
  \item  What is the determinant of $B$?  
  \item  What is the determinant of $C$?  
  \end{enumerate}

\item  
  \begin{enumerate}
  \item  Find an orthogonal basis for the column space of $A$.
  \item  Find an orthonormal basis for the column space of $A$.
  \end{enumerate}
  (Hint: be careful -- the column space of $A$ has dimension 2 and not dimension 3.)

\item  
  \begin{enumerate}
  \item  Find all eigenvalues of $A$.
  \item  Is $A$ diagonalizable? If so, explain why and what is the diagonal matrix?
    If not, explain why not.
    \\
    (If $A$ is diagonalizable, you don't need to find the diagonalzing matrix.)
  \end{enumerate}

\item  
  \begin{enumerate}
  \item  Find all eigenvalues of $C$.
  \item  For each eigenvalue of $C$, find a basis for the associated eigenspace.
  \item  If possible, diagonalize $C$ (find the diagonalizing matrix and the diagonal matrix).
    If this is not possible, explain why not.
  \end{enumerate}

\item  Suppose a $n \times n$ matrix $M$, has a null space of dimension $2$.
  Show that $M$ has an eigenvalue of $0$.
  \\
  (Hint: the dimension of $2$ doesn't matter, what matters is that the null space is nontrivial.)
  
\item  (Extra Credit, 5 points)
  If $A$ is diagonalizable, find a matrix $Q$ so that $Q^{-1}AQ$ is diagonal.
  If $A$ is not diagonalizable, find a matrix $Q$ so that $Q^{-1}AQ$ is triangular.


\end{enumerate}




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