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\lhead{{\sf Summer Bridge Course (Algebra)}: Friday July 24}
\chead{}\rhead{\thepage}
\lfoot{Due Monday, July 27 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
\pagestyle{fancy}

\begin{document}

%\(\;\) \vspace{-4mm}

\noindent \textbf{Topics}: Determinant and trace. \\

\noindent \textbf{Do}: Answer the following questions. \\
%%------------------------------------------------------------------------

%% HW 14, Problem 1
\begin{exercise}
  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---purely a geometric argument---find \(|\det\bold{H}|\). \\
\end{exercise}

%%------------------------------------------------------------------------

%% HW 14, Problem 2
\begin{exercise}
  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 simple to compute each individual determinant. Compute the determinant of each of the following matrices using this method. Only include the non-zero terms.
  \[
  \A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\qquad\qquad
  \B=\begin{bmatrix}1&1&0&0\\1&0&1&0\\0&0&1&1\\0&2&0&1\end{bmatrix}\,
  \qquad\qquad
  \mathbf{C}=\begin{bmatrix}1&0&0&2\\0&3&4&5\\5&4&0&3\\2&0&0&1\end{bmatrix}\,
  \]  
  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.''] \\
\end{exercise}
    
%\begin{solution}
  %%
  %% Here's how to make an "augmented matrix" in LaTeX:
  %  \[
  %    [\A\;|\;\I\,]
  %    =\begin{bmatrix}[cc|cc] 1 & -2 & 1 & 0 \\ 3 & -7 & 0 & 1\end{bmatrix}
  %    \,\qquad\qquad
  %    [\B\;|\;\I\,]=\begin{bmatrix}[ccc|ccc] 3 & 1 & 4 & 1 & 0 & 0 \\ 
  %    5 & 2 & 6 & 0 & 1 & 0 \\ -7 & -2 & -9 & 0 & 0 & 1\end{bmatrix}
  %  \]
%\end{solution}

 %%------------------------------------------------------------------------

%% HW 14, Problem 3
 \begin{exercise}
\item Recall that the \emph{trace} of an \(n\times n\) matrix is \(\tr A=\Sum_{i=1}^n a_{ii}\).
  \begin{enumerate}[label=(\roman*)]
  \item Let \(A\) be an \(m\times n\) matrix, and \(B\) be an \(n\times m\) matrix. Show that the \(m\times m\) matrix \(AB\) and the \(n\times n\) matrix \(BA\) have the same trace. 
  \item If \(A\) is an \(n\times n\) matrix, derive a formula for \(\tr(A^TA)=\tr(AA^T)\) in terms of \(a_{ij}\).
  \end{enumerate}
  \end{exercise}

%\begin{solution}
%\end{solution}

 %%------------------------------------------------------------------------ 

\end{document}
