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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due November 2 (Friday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 311 $|$ Linear Algebra $|$ Fall 2012 
    $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Strang, Section 5.1, 5.2, 5.3. \\

\noindent\emph{Suggested short conceptual exercises}: Strang, Section
5.1, \#1, 2, 4--6, 8, 11, 12, 17, 20, 28, 29. Section 5.2, 
\#5--10, 23. Section 5.3, \#4, 7, 9, 10, 14, 15, 21--23.


\begin{enumerate}

  % Strang: Section 5.1, #27 
\item Use elemenatry row operations to compute the determinants of the
  following matrices.
  \[
  \A=\begin{bmatrix}0&a&0\\0&0&b\\c&0&0\end{bmatrix}\,\qquad\qquad
  \B=\begin{bmatrix}0&a&0&0\\0&0&b&0\\0&0&0&c\\d&0&0&0\end{bmatrix}\,
  \qquad\qquad
  \C=\begin{bmatrix}a&a&a\\a&b&b\\a&b&c\end{bmatrix}\,.
  \]
  


  % Strang: Section 5.1, #22-23 (modified)
\item Recall that the determinant of a $2\times 2$ matrix is $ad-bc$.
  Carry out the steps outlined below for the matrix $\A$. Then carry
  them out for $\B$.
  \[
  \A=\begin{bmatrix}2&1\\1&2\end{bmatrix}\,\qquad\qquad\qquad
  \B=\begin{bmatrix}4&1\\2&3\end{bmatrix}\,.
  \]
  \begin{enumerate}
  \item Compute the determinant of three matrices: $\A$ and $\A^{-1}$ and
    $\A-\lambda\I$, where $\lambda$ is a fixed parameter. 
  \item Determine which two numbers $\lambda$ lead to $\det(\A-\lambda\I)=0$. 
  \item Write down the matrix $\A-\lambda\I$ for both of these
    values of $\lambda$. Note that these matrices should not be
    invertible.
  \end{enumerate}


  
  % Made up myself (one matrix is from Strang 5.2, #4)
\item Using linearity of each row, the determinant of an $n\times n$
  matrix can be written as a sum of determinants of no more than $n!$
  matrices that have \emph{exactly one non-zero entry in each row and
  column}. For example, a $3\times 3$ determinant breaks up as
  \[
  \begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}
  \end{vmatrix}  
  =\begin{vmatrix}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\end{vmatrix}  
  +\begin{vmatrix}a_{11}&0&0\\0&0&a_{23}\\0&a_{32}&0\end{vmatrix}
  +\begin{vmatrix}0&a_{12}&0\\a_{21}&0&0\\0&0&a_{33}\end{vmatrix}+\cdots.
  \]
  From here, it is easy to compute each individual determinant.
  Compute the determinant of each of the following matrices using this
  method. Only include the non-zero terms.
  \[
  \A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\qquad\qquad
  \B=\begin{bmatrix}1&1&0&0\\1&0&1&0\\0&0&1&1\\0&2&0&1\end{bmatrix}\,
  \qquad\qquad
  \C=\begin{bmatrix}1&0&0&2\\0&3&4&5\\5&4&0&3\\2&0&0&1\end{bmatrix}\,
  \]  
  Use your answer to the first part to derive a ``shortcut formula'' for
  the determinant of any $3\times 3$ matrix. [\emph{Hint}: Write out
    the augmented $3\times 6$ matrix $[\A|\A]$ and draw some
    ``diagonal lines.'']



  % Made up myself
\item Compute the determinants of the following matrices by \emph{cofactor
  expansion}:
  \[
  \A=\begin{bmatrix}1&-1&2\\1&3&4\\2&1&1\end{bmatrix}\qquad
  \B=\begin{bmatrix}1&2&0\\0&3&0\\0&7&1\end{bmatrix}\qquad
  \C=\begin{bmatrix}1&2&-7&4\\0&3&42&1\\0&0&2&0\\-1&3&9&-2\end{bmatrix}\,.
  \]
  To simplify your calculations, make a wise choice of which row or
  column to expand across. 



  % Strang: Section 5.2, #13 
\item The $n\times n$ determinant $C_n$ has $1$'s above and below the
  main diagonal:
  \[
  C_1=|0|\,,\qquad
  C_2=\begin{vmatrix}0&1\\1&0\end{vmatrix}\,,\qquad
  C_3=\begin{vmatrix}0&1&0\\1&0&1\\0&1&0\end{vmatrix}\,,\qquad
  C_4=\begin{vmatrix}0&1&0&0\\1&0&1&0\\0&1&0&1\\0&0&1&0\end{vmatrix}\,.
  \]
  \begin{enumerate}
    \item Use cofactor expansions to compute the determinants $C_1$,
      $C_2$, $C_3$, and $C_4$.
    \item Find the relation between $C_n$ and $C_{n-1}$ and
      $C_{n-2}$. Compute $C_{10}$.
  \end{enumerate}



  % Strang: Section 5.3, #8 (modified)
\item Let $\A=\begin{bmatrix}1&1&4\\1&2&2\\1&2&5\end{bmatrix}$.
  \begin{enumerate}
    \item Find the cofactors of $\A$, put them into the cofactor
      matrix $\C$.
    \item Use $\A$ and $\C$ to compute $\det\A$.
    \item Use Part~(b) to compute $\A^{-1}$.
    \item Suppose that the $4$ in $\A$ was changed to $100$. Which
      of $\C$, $\det\A$, and $\A^{-1}$ would change?
  \end{enumerate}



  % Strang: Section 5.3, #11-12
\item Suppose $\A$ is an $n\times n$ matrix with integer entries. 
  \begin{enumerate}
    \item Prove that if $\det\A=\pm 1$, then all entries of $\A^{-1}$
      are integers.
    \item Prove that if all entries of $\A^{-1}$ are integers, then
      $\det\A=\pm 1$.
  \end{enumerate}



  % Strang: Section 5.3, #20
\item The following matrix is called a ($4\times 4$) \emph{Hadamard matrix}:
  \[
  \bold{H}
  =\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\1&-1&-1&1\\1&-1&1&-1\end{bmatrix}\,.
  \]
  Note that the ``box'' formed by the four row (or column) vectors is a
  hypercube in $\R^4$. Using this information alone, compute
  $|\det\bold{H}|$.


\end{enumerate}


\end{document}



