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	\title{MATH 117: Daily Assignment 8}
	
	\author{WRITE YOUR NAME HERE}
	
	\maketitle
	
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	 See the \href{https://people.ucsc.edu/~jbreland/teaching/SM22_MATH117/daily_assignments.html}{daily assignment webpage} for due dates, templates, and assignment description.
	Try to explain your reasoning and justify your computations for every problem. You should not appeal to any theorems that we have not proved yet. 
	
	%The enumerate enviroment produces numbered lists. The command \item specifies a list item.
	\begin{enumerate}[\bfseries 1.]
		
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		\item 
		\begin{enumerate}[(a)]
			\item Compute the determinant of $A = \begin{pmatrix}
		1 & 1 & 3\\ 1 & 2 & 3\\
		3 & 1 & 0
		\end{pmatrix} \in (\Z_5)^{3\times 3}$ using the Liebniz formula.
	
	\item Compute the determinant of $B = \begin{pmatrix} 
			1 & 3 & 4\\ 2 & 3 & 4\\ 1 & 0 & 1	
			\end{pmatrix} \in (\Z_5)^{3\times 3}$ using row operations. 
		
		\item Compute the determinant of $C = \begin{pmatrix}
			2 & 0 & 0 & 2\\
			1 & 1 & 3 & 4\\
			1 & 2 & 3 & 4\\
			3 & 1 & 0 & 1
		\end{pmatrix}$ using \href{https://en.wikipedia.org/wiki/Laplace_expansion}{Cofactor expansion} along the first row.
	
	
			
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		\item Let $F = \Z_2$ and let $V=F^2$. Let $L_A:V \to V$ be the linear map given by left multiplication with the matrix $A = \begin{pmatrix}
			1 & 1\\  0 & 1
		\end{pmatrix}$. Similarly, defined $L_B:V\to V$ via left multiplication with the matrix $B = \begin{pmatrix}
		1 & 0\\  1 & 1
	\end{pmatrix}$. 
	\begin{enumerate}[(a)]
		\item Compute $\mathrm{cp}_{L_A \otimes L_B}(x)$. You may use the "Cofactor expansion" algorithm, even though we have not proved it. 
		
		\item Find the eigenvalues of $L_A \otimes L_B$
		
		\item Find a basis for each eigenspace of $L_A \otimes L_B$.
		
	\end{enumerate}
		
	
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		\item Let $F = \Z_2$ and $V = F^2$. Find all linear operators $L:V \to V$ for which $\mathrm{cp}_L(x) = x^2 + 1$. 

		
	
		
		
		

		
		

				
			
		
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