Virtual Math Learning Center

1. Consider the matrices \(A, B, C, D, E\) and \(G\) defined below. 
\[A=\left[\begin{array}{ccc} 1 & 0 & -y\\ 5 & x & 11\end{array}\right]
\quad
B=\left[\begin{array}{cccc}
1 & 2 & 3 & 4\\
5 & 6 & 7 & 8\\
9 & 10 & 11 & 12
\end{array}
\right]
\]
\[
C=\left[\begin{array}{cccc}
u & v & 5 & 3\\
-1 & 8 & 7 & 0
\end{array}\right]
\quad 
D=\left[\begin{array}{ccc}
5 & 0 & 4
\end{array}\right]
\]
\[
E=\left[\begin{array}{cc}
8 & 0\\
1 & -5\\
w & 6
\end{array}\right]
\quad 
G=\left[\begin{array}{cccc} 
6 & 12 & t & 3\\
-1 & 8 & 7 & p
\end{array}\right]
\]
Find the dimensions of the matrices defined in parts a–d.

1. \(B\)
2. \(C^T\)
3. \(D\)
4. \(\left(E^T\right)^T\)

2. Consider the matrices \(A, B, C, D, E\) and \(G\) defined below. 
\[A=\left[\begin{array}{ccc} 1 & 0 & -y\\ 5 & x & 11\end{array}\right]
\quad
B=\left[\begin{array}{cccc}
1 & 2 & 3 & 4\\
5 & 6 & 7 & 8\\
9 & 10 & 11 & 12
\end{array}
\right]
\]
\[
C=\left[\begin{array}{cccc}
u & v & 5 & 3\\
-1 & 8 & 7 & 0
\end{array}\right]
\quad 
D=\left[\begin{array}{ccc}
5 & 0 & 4
\end{array}\right]
\]
\[
E=\left[\begin{array}{cc}
8 & 0\\
1 & -5\\
w & 6
\end{array}\right]
\quad 
G=\left[\begin{array}{cccc} 
6 & 12 & t & 3\\
-1 & 8 & 7 & p
\end{array}\right]
\]
Find the value of each entry, if it exists

1. \(b_{23}\)
2. \(b_{32}\)

3. Consider the matrices \(A, B, C, D, E\) and \(G\) defined below. 
\[A=\left[\begin{array}{ccc} 1 & 0 & -y\\ 5 & x & 11\end{array}\right]
\quad
B=\left[\begin{array}{cccc}
1 & 2 & 3 & 4\\
5 & 6 & 7 & 8\\
9 & 10 & 11 & 12
\end{array}
\right]
\]
\[
C=\left[\begin{array}{cccc}
u & v & 5 & 3\\
-1 & 8 & 7 & 0
\end{array}\right]
\quad 
D=\left[\begin{array}{ccc}
5 & 0 & 4
\end{array}\right]
\]
\[
E=\left[\begin{array}{cc}
8 & 0\\
1 & -5\\
w & 6
\end{array}\right]
\quad 
G=\left[\begin{array}{cccc} 
6 & 12 & t & 3\\
-1 & 8 & 7 & p
\end{array}\right]
\]
Find the value of each entry, if it exists.

1. \(c_{24}+4b_{32}\)
2. \(f_{13}\) given \(F=A+E^T\)

4. Consider the matrices \(A, B, C, D, E\) and \(G\) defined below. 
\[A=\left[\begin{array}{ccc} 1 & 0 & -y\\ 5 & x & 11\end{array}\right]
\quad
B=\left[\begin{array}{cccc}
1 & 2 & 3 & 4\\
5 & 6 & 7 & 8\\
9 & 10 & 11 & 12
\end{array}
\right]
\]
\[
C=\left[\begin{array}{cccc}
u & v & 5 & 3\\
-1 & 8 & 7 & 0
\end{array}\right]
\quad 
D=\left[\begin{array}{ccc}
5 & 0 & 4
\end{array}\right]
\]
\[
E=\left[\begin{array}{cc}
8 & 0\\
1 & -5\\
w & 6
\end{array}\right]
\quad 
G=\left[\begin{array}{cccc} 
6 & 12 & t & 3\\
-1 & 8 & 7 & p
\end{array}\right]
\]
Compute each of the matrices defined below, if such matrices are defined.

1. \(\dfrac{1}{2}E\)
2. \(\dfrac{1}{2}E+D\)
3. \(2G-4C\)

5. Consider the matrices \(A, B, C, D, E\) and \(G\) defined below. 
\[A=\left[\begin{array}{ccc} 1 & 0 & -y\\ 5 & x & 11\end{array}\right]
\quad
B=\left[\begin{array}{cccc}
1 & 2 & 3 & 4\\
5 & 6 & 7 & 8\\
9 & 10 & 11 & 12
\end{array}
\right]
\]
\[
C=\left[\begin{array}{cccc}
u & v & 5 & 3\\
-1 & 8 & 7 & 0
\end{array}\right]
\quad 
D=\left[\begin{array}{ccc}
5 & 0 & 4
\end{array}\right]
\]
\[
E=\left[\begin{array}{cc}
8 & 0\\
1 & -5\\
w & 6
\end{array}\right]
\quad 
G=\left[\begin{array}{cccc} 
6 & 12 & t & 3\\
-1 & 8 & 7 & p
\end{array}\right]
\]
Find the values of \(p, t, u,\) and \(v\) that will make \(G = C.\)
Note: In the video, matrix G is called matrix B.

6. The Bryan-College Station area has 3 different locations of Jay’s restaurants; Wellborn Road, Walton Drive, and Wayfair Circle. At closing time on Thursday evening the manager at each location logged the inventory of certain items. The manager at the Wellborn Road location noted her store had 55 containers of potato salad, 60 pounds of chicken fingers, 100 loaves of Texas toast, and 75 bags of seasoned fries. The manager at the Walton Drive location noted they had 20 containers of potato salad and 65 loaves of Texas toast, but no chicken fingers or seasoned fries. The Wayfair Circle manager noted he had 95 pounds of chicken fingers and 45 bags of seasoned fries, but no containers of potato salad and no loaves of Texas toast.

1. Create a \(4 \times 3\) matrix, \(F\) , to represent the inventory of these items at closing time on Thursday at each location of Jay’s restaurants. Clearly label your rows and columns of \(F\).
2. Let matrix \(D\) (below) be the desired amount of potato salad, chicken fingers, Texas toast, and seasoned fries for each location when it opens for business on Friday. Define matrix \(S\) as \(S = D − F\). Find \(S\) and determine what this matrix represents in the context of this problem. \[D=\begin{array}{lc}
    & \begin{array}{ccc} \textnormal{Wellborn Rd.} \; & \textnormal{Walton Dr.}\; & \textnormal{Wayfair Cr.}\end{array}\\
    \begin{array}{l}\textnormal{Potato Salad} \\ \textnormal{Chicken Fingers}\\ \textnormal{Tx Toast}\\ \textnormal{Seasoned Fries}\\
    \end{array}
    & 
    \left[\begin{array}{ccc} 
    \hspace{2.5em}100\hspace{2.5em}  & \hspace{2.5em}75\hspace{2.5em} & \hspace{2.5em}88\hspace{2.5em} \\
    125 & 110 & 150 \\
    160 & 125 & 135 \\
    105 & 115 & 150
    \end{array}\right]\end{array}\]
3. There is an Aggie Football game in the area on Saturday and each store manager expects a \(35\%\) increase in the amount of each item in matrix \(D\). Write a matrix equation that represents the amount of potato salad, chicken fingers, Texas toast, and seasoned fries each location will need to order when it opens for business on Saturday.