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	\title{MATH 117: Daily Assignment 9}
	
	\author{WRITE YOUR NAME HERE}
	
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	 See the \href{https://people.ucsc.edu/~jbreland/teaching/SM23_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. 
	
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		\item Read Section 3.4 on your own. You can skip the reading if you are already familiar with the symmetric group. Write each permutation below as a product of transpositions. Then compute the sign of each permutation. 
		\begin{enumerate}[(a)]
			\item $(1,3,4,2) \in S_7$ 
			\item $(1,4,5,7,6) \in S_7$
			\item $ (1,2)(3,4,5,7,6) \in S_7$.
		\end{enumerate}
		
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		\item 
		\begin{enumerate}[(a)]
			\item Compute the determinant of $A = \begin{psmallmatrix}
				1 & 1 & 3\\ 1 & 2 & 3\\
				3 & 1 & 0
			\end{psmallmatrix} \in (\Z_5)^{3\times 3}$ using the Liebniz formula.
			
			\item Compute the determinant of $B = \begin{psmallmatrix} 
				1 & 3 & 4\\ 2 & 3 & 4\\ 1 & 0 & 1	
			\end{psmallmatrix} \in (\Z_5)^{3\times 3}$ using row operations. 
			
			\item Compute the determinant of $C = \begin{psmallmatrix}
				2 & 0 & 0 & 2\\
				1 & 1 & 3 & 4\\
				1 & 2 & 3 & 4\\
				3 & 1 & 0 & 1
			\end{psmallmatrix}$ using \href{https://en.wikipedia.org/wiki/Laplace_expansion}{Cofactor expansion}\footnote{I won't prove this because it takes too long. Feel free to use it whenever you need to compute determinants.} along the first row.
		
		\item Compute the determinant of 
		\[
		D = \begin{psmallmatrix}
			1 & 1 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
			1 & 2 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
			3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
			 0 & 0 & 0 &1 & 3 & 4 & 0 & 0 & 0 & 0\\
			 0 & 0 & 0 &2 & 3 & 4 & 0 & 0 & 0 & 0\\ 
			 0 & 0 & 0 &1 & 0 & 1 & 0 & 0 & 0 & 0\\
			 0 & 0 & 0 & 0 & 0 & 0 &2 & 0 & 0 & 2\\
			 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 3 & 4\\
			 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 3 & 4\\
			 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 & 0 & 1
		\end {psmallmatrix}
		\]
		without doing any additional work. 
			
			\item Which matrices above are invertible, if any?
			
		\end{enumerate}
		
		
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		\item Repeat Weekly 3.4(ii) using Proposition 3.3.10. Did you compute the matrix correctly the first time?
		
		
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