\begin{questions}
%1
\begin{question} 
Consider the following matrices:
$$A = \left[\begin{array}{cc}5 & w\\2 & 0\end{array}\right], \qquad B= \left[\begin{array}{cc}7 & 4\\x & 1\end{array}\right], \qquad C = \left[\begin{array}{ccc}y & 2 & 6\\1 & z & 7\end{array}\right].$$
Which of the following matrix operations \textbf{cannot} be performed?

\begin{choices}
\choice[!]    {All of the below operations can be performed.}
\choice        {\( 9C \)}
\choice        {\( AB \)}
\choice        {\( BC \)}
\choice        {\( A - B \)}
\end{choices}
\end{question}


%2
\begin{question} 
Consider the following matrices:
$$D = \left[\begin{array}{r}7 \\ 2 \end{array}\right], \qquad E = \left[\begin{array}{r}y \\ z\end{array}\right].$$
Determine the matrix $5D + E$.

\begin{choices}
\choice        {\( \left[\begin{array}{rr}35 & y\\10 & z\end{array}\right]\)} %2x2 matrix with columns 5D & E
\choice        {\( \left[\begin{array}{r}7 + 5y\\2 + 5z\end{array}\right]\)} %D + 5E
\choice[!]    {\( \left[\begin{array}{r}35 + y\\10 + z\end{array}\right]\)}
\choice        {\( \left[\begin{array}{rr}35 + 5y\\10 + 5z\end{array}\right]\)} %5(D + E)
\choice        {\( \left[\begin{array}{r}45 + y + z\end{array}\right]\)} %1x1 matrix adding all entries of 5D and E
\end{choices}
\end{question}


\NewPage
%3
\begin{question} 
Consider the following matrices:
$$X = \left[\begin{array}{rr}b & 7 \\ -2 & 3\end{array}\right], \qquad Y = \left[\begin{array}{rr}9 & -4 \\ a & 0\end{array}\right].$$
Determine the matrix $YX$.

\begin{choices}
\choice        {\( \left[\begin{array}{rr}9 + b & 3\\a - 2 & 3\end{array}\right]\)} %Y + X
\choice        {\( \left[\begin{array}{rr}1 & 0 \\ 0 & 1 \end{array}\right]\)} %identity
\choice        {\( \left[\begin{array}{rr}9b & -28\\-2a & 0\end{array}\right]\)} %multiply corresponding entries
\choice        {\( \left[\begin{array}{rr}9b + 7a & -4b\\-18 + 3a & 8\end{array}\right]\)} %XY
\choice[!]    {\( \left[\begin{array}{rr}9b + 8 & 51\\ab & 7a\end{array}\right]\)}
\end{choices}
\end{question}


%4
\begin{question} 
Suppose $R$, $S$ and $T$ are matrices such that the product $RST$ is a $5 \times 11$ matrix and $S$ is a $10 \times 6$ matrix.  What are the dimensions of $R$?

\begin{choices}
\choice        {\(R \text{ is a } 5 \times 6 \text{ matrix}\)} %dimensions of RS
\choice        {\(R \text{ is a } 10 \times 11 \text{ matrix}\)} %dimensions of ST
\choice        {\(R \text{ is a } 6 \times 11 \text{ matrix}\)} %dimensions of T
\choice[!]    {\(R \text{ is a } 5 \times 10 \text{ matrix}\)}
\choice        {\(R \text{ is a } 10 \times 6 \text{ matrix}\)} %dimensions of S
\end{choices}
\end{question}


%5
\begin{question} 
A linear system has 5 equations and 8 variables.  How many rows and columns does the corresponding augmented matrix have?

\begin{choices}
\choice[!]    {5 rows and 9 columns}
\choice       {6 rows and 8 columns} %#rows=#equations+1 and #columns=#variables
\choice       {5 rows and 8 columns} %#rows=#equations and #columns=#variables
\choice       {6 rows and 9 columns} %#rows=#equations+1 and #columns=#variables+1
\choice       {5 rows and 7 columns} %#rows=#equations and #columns=#variables-1
\end{choices}
\end{question}


\NewPage
%6
\begin{question} 
Which matrix results from applying the row operation $R_1 \to R_1 + 3R_2$ to the matrix below?
$$\left[ \begin{array}{rrr|r}
1 & -3 & 5 & 6\\
0 & 1 & -4 & 1\\
0 & -5 & 7 & -9
\end{array} \right]$$

\begin{choices}
\choice[!]    {\(\left[ \begin{array}{rrr|r}1 & 0 & -7 & 9\\0 & 1 & -4 & 1\\0 & -5 & 7 & -9\end{array} \right]\)}
\choice       {\(\left[ \begin{array}{rrr|r}1 & 0 & 5 & 6\\0 & 1 & -4 & 1\\0 & -5 & 7 & -9\end{array} \right]\)} %only operates on the first two entries of row 1
\choice       {\(\left[ \begin{array}{rrr|r}1 & 0 & 8 & 9\\0 & 1 & -4 & 1\\0 & -5 & 7 & -9\end{array} \right]\)} %adds 3 to the last three entries of row 1
\choice       {\(\left[ \begin{array}{rrr|r}1 & 0 & -7 & 6\\0 & 1 & -4 & 1\\0 & -5 & 7 & -9\end{array} \right]\)} %only operates on the first three entries of row 1
\choice       {\(\left[ \begin{array}{rrr|r}1 & -3 & 5 & 6\\0 & 3 & -12 & 3\\0 & 0 & 0 & 0\end{array} \right]\)} %R_1 \to R_1, R_2 \to 3R_2, R_3 is all zeros
\end{choices}
\end{question}


%7
\begin{question} 
Which operation is a valid row operation that can be applied to the following augmented matrix to change the boxed entry to 1?
$$\left[ \begin{array}{rrc|r}
1 & 0 & 5 & 16\\
0 & 1 & 7 & 29\\
0 & 0 & \fbox{5} & -32
\end{array} \right]$$

\begin{choices}
\choice       {\(R_3 \to R_3 - 4\)} %subtract a constant
\choice[!]    {\(R_3 \to \frac{1}{5}R_3\)}
\choice       {\(R_3 \to \frac{R_3}{R_1}\)} %divide rows
\choice       {\(R_3 \to \frac{R_3}{R_3}\)} %divide row by itself
\choice       {\(R_3 \to R_3 + \frac{1}{5}\)} %add (instead of multiply) by scalar
\end{choices}
\end{question}


\NewPage
%8
\begin{question} 
The Gauss-Jordan method was applied to a system of linear equations to obtain the following augmented matrix in reduced row echelon form.
$$\begin{blockarray}{rrrr}
x & y & z & \\
\begin{block}{[rrr|r]}
\phantom{1}1 & 0 & 0 & -57\phantom{1}\bigstrut[t]\\
\phantom{1}0 & 1 & 0 & 34\phantom{1}\\
\phantom{1}0 & 0 & 1 & -8\phantom{1}\\
\phantom{1}0 & 0 & 0 & 0\phantom{1}\bigstrut[b]\\
\end{block}
\end{blockarray}\vspace{-\baselineskip}$$
How many solutions does the original system of equations have?

\begin{choices}
\choice[!]    {Exactly 1 solution}
\choice       {Exactly 2 solutions}
\choice       {Exactly 3 solutions}
\choice       {No solution}
\choice       {Infinitely many solutions}
\end{choices}
\end{question}


%9
\begin{question} 
The following augmented matrix is almost in reduced row echelon form.   What is the solution to the corresponding system of linear equations?
$$\begin{blockarray}{rrrrr}
w & x & y & z & \\
\begin{block}{[rrrr|r]}
\phantom{1}0 & 1 & 0 & 0 & -39\phantom{1}\bigstrut[t]\\
\phantom{1}1 & 0 & 0 & 0 & 72\phantom{1}\\
\phantom{1}0 & 0 & 1 & 0 & -68\phantom{1}\\
\phantom{1}0 & 0 & 0 & 1 & 54\phantom{1}\bigstrut[b]\\
\end{block}
\end{blockarray}\vspace{-\baselineskip}$$

\begin{choices}
\choice       {\((w,x,y,z) = (0, 0, -68, 54)\)} %only use pivots in correct position
\choice       {\((w,x,y,z) = (-72, 39, 68, -54)\)} %negative of answer
\choice       {\((w,x,y,z) = (-39, -68, 54, 72)\)} %(x,y,z,w)
\choice[!]    {\((w,x,y,z) = (72, -39, -68, 54)\)}
\choice       {\((w,x,y,z) = (-39, 72, -68, 54)\)} %order from matrix
\end{choices}
\end{question}


\NewPage
%10
\begin{question} 
The following augmented matrix is almost in reduced row echelon form.   What is the solution to the corresponding system of linear equations?
$$\begin{blockarray}{rrrr}
x & y & z & \\
\begin{block}{[rrr|r]}
\phantom{1}1 & 0 & 4 & 83\phantom{1} \bigstrut[t]\\
\phantom{1}0 & 1 & -9 & 56\phantom{1}\\
\phantom{1}0 & 0 & 0 & 1\phantom{1}\bigstrut[b]\\
\end{block}
\end{blockarray}\vspace{-\baselineskip}$$

\begin{choices}
\choice[!]    {The system of equations has no solution.}
\choice       {\((x,y,z) = (83, 56, 1)\)} %RHS column
\choice       {\((x,y,z) = (79, 65, 1)\)} %Subtract z column from RHS
\choice       {\((x,y,z) = (83 - 4z, 56 + 9z, 1)\)} %infinitely many solutions
\choice       {\((x,y,z) = (83 + 4z, 56 - 9z, 1)\)} %infinitely many solutions with wrong signs
\end{choices}
\end{question}


%11
\begin{question} 
The following augmented matrix is in reduced row echelon form.   What is the solution to the corresponding system of linear equations?
$$\begin{blockarray}{rrrrr}
w & x & y & z & \\
\begin{block}{[rrrr|r]}
\phantom{1}1 & 0 & 0 & 0 & -8\phantom{1} \bigstrut[t]\\
\phantom{1}0 & 1 & 0 & -2 & 5\phantom{1}\\
\phantom{1}0 & 0 & 1 & 9 & 4\phantom{1}\bigstrut[b]\\
\end{block}
\end{blockarray}\vspace{-\baselineskip}$$

\begin{choices}
\choice       {The system of equations has no solution.}
\choice       {\((w,x,y,z) = (-8, 5, 4, 7)\)} %RHS column and add z column
\choice       {\((w,x,y,z) = (-8, 7, -5, 0)\)} %Subtract z column from RHS
\choice       {\((w,x,y,z) = (-8, 5 - 2z, 4 + 9z, z)\)} %infinitely many solutions with wrong signs
\choice[!]    {\((w,x,y,z) = (-8, 5 + 2z, 4 - 9z, z)\)}
\end{choices}
\end{question}


\NewPage
%12
\begin{question} 
Determine all values of $k$ for which the following matrix is invertible.
$$\left[ \begin{array}{rr}
2 & 8\\
k & 9
\end{array} \right]$$

\begin{choices}
\choice       {The matrix is invertible for all values of $k$.}
\choice       {\( k = -2.25 \)} %not invertible using determinant = ad + bc
\choice       {\( k = 2.25 \)} %not invertible
\choice       {\( k \neq -2.25 \)} %invertible using determinant = ad + bc
\choice[!]    {\( k \neq 2.25 \)}
\end{choices}
\end{question}


%13
\begin{question} 
Let
$$A = \left[ \begin{array}{rr}
2 & 4\\
8 & 6
\end{array} \right].$$
Determine the sum $A + I_2$, where $I_2$ is the $2 \times 2$ identity matrix.

\begin{choices}
\choice       {\(\left[ \begin{array}{rr}3 & 5\\9 & 7\end{array} \right]\)} %add 1 to all entries
\choice[!]    {\(\left[ \begin{array}{rr}3 & 4\\8 & 7\end{array} \right]\)}
\choice       {\(\left[ \begin{array}{rr}2 & 4\\8 & 6\end{array} \right]\)} %AI
\choice       {\(\left[ \begin{array}{rr}4 & 4\\8 & 8\end{array} \right]\)} %add 2 to diagonal entries
\choice       {\(\left[ \begin{array}{rr}2 & 5\\9 & 6\end{array} \right]\)} %add 1 to off diagonal entries
\end{choices}
\end{question}


\NewPage
%14
\begin{question} 
Consider the matrix equation
$$A\left[\begin{array}{r}x \\ y\end{array}\right] = \left[\begin{array}{r}-4 \\ 9\end{array}\right],$$
where $A^{-1} = \left[\begin{array}{rr}2 & 5 \\ 3 & 8\end{array}\right]$.  Determine the solution $\left[\begin{array}{r}x \\ y\end{array}\right]$.

\begin{choices}
\choice        {\(\left[\begin{array}{r}x \\ y\end{array}\right] = \left[\begin{array}{r}-77\\30\end{array}\right]\)} %AB
\choice        {\(\left[\begin{array}{r}x \\ y\end{array}\right] = \left[\begin{array}{r}19\\52\end{array}\right]\)} %columns of A times B
\choice        {\(\left[\begin{array}{r}x \\ y\end{array}\right] = \left[\begin{array}{r}-59\\38\end{array}\right]\)} %columns of A^-1 times B
\choice        {\(\left[\begin{array}{r}x \\ y\end{array}\right] = \left[\begin{array}{r}-4\\9\end{array}\right]\)} %B
\choice[!]    {\(\left[\begin{array}{r}x \\ y\end{array}\right] = \left[\begin{array}{r}37\\60\end{array}\right]\)}
\end{choices}
\end{question}


%15
\begin{question} 
A UK gift shop sells hats, jerseys and t-shirts.  A hat costs \$15, a jersey costs \$80, and a t-shirt costs \$20.  On a given weekend, 6 times the number of jerseys sold was the same as the number of t-shirts sold.  The store brought in a total of \$2{,}000 that weekend, and the money made from t-shirts alone was \$240 more than the money made from hats alone.  Let $x$ denote the number of hats sold, $y$ denote the number of jerseys sold, and $z$ denote the number of t-shirts sold.

\noindent Which one of the following equations \textbf{should not} be used to determine the number of hats, jerseys, and t-shirts sold that weekend?

\begin{choices}
\choice        {\( 15x + 80y + 20z = 2000 \)}
\choice        {\( 20z = 15x + 240 \)}
\choice[!]    {\( x + y + z = 115 \)}
\choice        {\( 6y = z \)}
\choice        {All of the above should be used to solve the system.}
\end{choices}
\end{question}


\NewPage
%16
\begin{question} A closed economy with two sectors has the following consumption matrix, $A$.
$$A = \left[\begin{array}{rr}0.3 & 0.9\\0.7 & 0.1\end{array}\right]$$
Assume the first row/column of $A$ corresponds to Sector 1 and the second row/column of $A$ corresponds to Sector 2.  

\noindent What percentage of Sector 1's production does Sector 2 consume?

\begin{choices}
\choice        {\( 90\% \)}
\choice        {\( 10\% \)}
\choice        {\( 50\% \)}
\choice        {\( 30\% \)}
\choice[!]    {\( 70\% \)}
\end{choices}
\end{question}


\NewPage
%17
\begin{center}
\hspace{-1.1cm}\textbf{Show all work to receive credit for the following problem.}
\end{center}

\vspace{-1em}

\QuestionSeparator

\begin{question} 
(10 points) Use the Gauss-Jordan method to determine the inverse of the matrix
$$A = \left[\begin{array}{rrr}1 & 2 & 1\\0 & -1 & 3\\-2 & -5 & 2\end{array}\right].$$
\textbf{Make sure to clearly mark your final answer.}
\end{question}


\NewPage
%18
\begin{center}
\hspace{-1.1cm}\textbf{Show all work to receive credit for the following problem.}
\end{center}

\vspace{-1em}

\QuestionSeparator

\begin{question} 
(10 points) A closed economy consists of two sectors: blacksmiths and miners.  The miners consume 40\% of their own production, and the blacksmiths consume the other 60\%.  The blacksmiths consume 20\% of their own production, and the miners consume the other 80\%.  Below is the consumption matrix, $A$.
$$A = \left[\begin{array}{rr}0.4 & 0.8\\0.6 & 0.2\end{array}\right]$$

(a) Write the associated augmented matrix $\left[\begin{array}{r|r}I_2 - A & \vec{0}\end{array}\right]$.\vfill

(b) Use between 2 and 4 row operations to put the matrix from part (a) in reduced row echelon form. \vfill \vfill \vfill

(c) How much must the miners produce for every \$15 produced by the blacksmiths?
\end{question}


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\end{questions}