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Problems PreCal 1508 PLTL Workshop, October 19, 2011\\
PLs: Alex Knaust, Edith Mejia. Lecturer: Yi-Yu Liao\\

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Please do the problems that you feel will help your group the most first (you don't have to do them in order). All handouts are available at \url{http://alex.knaust.info/pltlfall2011/}

\begin{enumerate}

\item Solve this system of equations using the inverse matrix.
\[\begin{bmatrix} 1 & 3 \\
0 & 2\end{bmatrix} \cdot 
\begin{bmatrix} x\\y \end{bmatrix}
 = \begin{bmatrix}4 \\1 \end{bmatrix}
 \]
 
\item Find the matrix $A$ if 
\[A = \begin{bmatrix}x & 2 \\ 6 & y\end{bmatrix}\quad\quad\quad A^2 = \begin{bmatrix}13 & 8 \\ 24 & 37\end{bmatrix}\]

\item Lets discover some useful properties of the determinant function of matricies
\begin{enumerate}
\item Find the determinant of $ A = \begin{bmatrix}2 & 1 \\ -3 & 5\end{bmatrix}$
\item What is the determinant of $A^{-1}$ and how is it related to the determinant of $A$?
\item Show why $\text{det}(k\cdot M) = k^2 \cdot \text{det}(M)$ if $M$ is a $2\times 2$ matrix and $k$ is some real number?
\item Is $\text{det}(Q\cdot R) = \text{det}(Q) \cdot \text{det}(R)$ for two $2\times 2$ matricies $Q$ and $R$?
\item Is $\text{det}(Q + R) = \text{det}(Q) + \text{det}(R)$ ?
\end{enumerate}

\item You have found at least two ways to solve systems of equations, one by performing row-operations on an augmented matrix, and another by multiplying by the inverse of the coefficient matrix. Why do they both work?

 I have the augmented matrix $A_0 = \left[\begin{array}{cc|c}1 & 2 & 0\\3 & 4 & 0\end{array}\right]$ and perform the Row-operation $R_2 = -3R_1 + R_2$ to get the augmented matrix $A_1 = \left[\begin{array}{cc|c}1 & 2 & 0\\0 & -2 & 0\end{array}\right]$
\begin{enumerate}
\item How could I have written this equation as $A\cdot X = B$? ($X$ is a matrix of unknowns...)
\item How can I describe the row operation as multiplication with a matrix? (Which matrix $C$ must I multiply by $A$ to get to $A_1$?)
\item Can all row operations be represented this way?
\item If they can, can you explain how using Gau\ss-Jordan elimination gives you the same result as multiplying by the inverse?
\end{enumerate}
\end{enumerate}
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