Exam 7: Matrices and Determinants

Write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables x, y, and z.) $\left[ \begin{array} { c c c : c } 1 & 8 & 9 & - 33 \\0 & 1 & - 1 & 2 \\0 & 0 & 1 & - 3\end{array} \right]$

Find all (a) minors and (b) cofactors of the matrix $\left[ \begin{array} { c c c } - 3 & 2 & - 8 \\3 & - 2 & 6 \\- 1 & 3 & - 6\end{array} \right]$ . Show your work.

Find the uncoded 1 × 3 row matrices for the message "MERRY CHRISTMAS"; then encode the message using the encoding matrix $\left[ \begin{array} { c c c } 0 & 2 & 1 \\2 & 1 & - 2 \\- 2 & - 2 & 0\end{array} \right]$ . Show all your work.

Determine whether the matrix is in row-echelon form and/or in reduced row-echelon form. $\left[ \begin{array} { l l l l } 1 & 0 & 5 & 9 \\0 & 1 & 0 & 6 \\0 & 0 & 1 & 0\end{array} \right]$

Use the matrix capabilities of a graphing utility to find AB, if possible. $A = \left[ \begin{array} { c c c } - 5 & 1 & - 9 \\5 & 5 & 7 \\9 & 5 & 1\end{array} \right] , B = \left[ \begin{array} { c c c } 7 & 2 & - 9 \\- 1 & - 7 & - 7 \\1 & - 7 & 7\end{array} \right]$

Evaluate the expression. $\left[ \begin{array} { c c } - 1 & - 8 \\2 & 5\end{array} \right] + \left[ \begin{array} { c c } - 9 & - 2 \\- 3 & - 1\end{array} \right] + \left[ \begin{array} { c c } 9 & 5 \\- 1 & - 9\end{array} \right]$

Find $A + B$ $A = \left[ \begin{array} { c c c } - 2 & - 1 & 6 \\1 & 6 & 2\end{array} \right]$ $B = \left[ \begin{array} { l l l } 3 & 0 & - 4 \\8 & 4 & - 5\end{array} \right]$

Find x and y. $\left[ \begin{array} { c c c } x - 6 & 8 & 5 y \\1 & 2 x & 9 \\9 & y + 8 & 1\end{array} \right] = \left[ \begin{array} { c c c } - 8 x - 42 & 8 & 15 \\1 & - 8 & 9 \\9 & 11 & 1\end{array} \right]$

Write the system of linear equations as a matrix equation $A X = B$ and then use Gauss-Jordan elimination on the augmented matrix $[ A : B ]$ to solve for the the matrix $X$ $\left\{ \begin{array} { c } - 5 x - 5 y + 9 z = 9 \\- 5 x - 5 y + 8 z = 8 \\3 x + 7 y + 2 z = - 10\end{array} \right.$

Find the determinant of the matrix $\left[ \begin{array} { l l } - 8 & 2 \\- 6 & 4\end{array} \right]$ .

Find the uncoded 1 × 3 row matrices for the message "TWO IF BY LAND"; then encode the message using the encoding matrix $\left[ \begin{array} { c c c } 0 & 4 & - 4 \\- 4 & - 1 & 1 \\- 1 & 4 & 0\end{array} \right]$ . Show all your work.

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $\left[ \begin{array} { c c c c } 1 & - 9 & 5 & 2 \\0 & 1 & 0 & - 9 \\0 & 0 & 1 & - 5\end{array} \right]$

A city zoo borrowed $2,000,000 at simple annual interest to construct a breeding facility. Some of the money was borrowed at 3%, some at 5%, and some at 10%. Use a system of equations to determine how much was borrowed at 5% if the total annual interest was $105,000 and the amount borrowed at 3% was twice the amount borrowed at 10%. Solve the system using matrices.

Consider the circuit shown in the figure below. The currents $I _ { 1 } , I _ { 2 } , \text { and } I _ { 3 }$ ( in amperes ) are the solutions to the system of linear equations $\left\{ \begin{aligned}3 I _ { 1 } + 6 I _ { 3 } & = E _ { 1 } \\2 I _ { 2 } + 6 I _ { 3 } & = E _ { 2 } \\I _ { 1 } + I _ { 2 } - I _ { 3 } & = 0\end{aligned} \right.$ , where $E _ { 1 } \text { and } E _ { 2 }$ are voltages of 6 and 9 volts, respectively. Use the inverse of the coefficient matrix of this system to find the unknown currents. Round answers to nearest tenth. [Hint: If a current value turns out to be negative, it simply means that the current flow is in the opposite direction from that indicated in the figure below.]

If possible, find AB. $A = \left[ \begin{array} { c c } 8 & 9 \\- 4 & 9 \\- 6 & 0\end{array} \right] , B = \left[ \begin{array} { c } - 6 \\- 1\end{array} \right]$

Find the determinant of $\left[ \begin{array} { c c c } - 9 & 3 & 0 \\6 & - 6 & 0 \\- 3 & - 24 & - 3\end{array} \right]$ .

A fruit grower raises apples and peaches, which are shipped to three different outlets. The number of units of each fruit that is shipped to each outlet is shown in the matrix $A$ $A = \left[ \begin{array} { l l l } 125 & 125 & 125 \\150 & 75 & 150\end{array} \right]$ where the first row contains the data for the number of apples shipped, the second row contains the data for the number of peaches shipped, and the columns correspond to outlets X, Y, and Z respectively. The profit per unit of apples is $4.50 and the profit per unit of peaches is $5.00. Organize the profits per unit in a matrix $B.$ $B = \left[ \begin{array} { l l } 4.50 & 5.00\end{array} \right]$ Compute $B A$ to find the profits from both crops at outlet $Y.$

Find the determinant of $\left[ \begin{array} { c c c } 0 & 4 & 0 \\6 & - 4 & 12 \\- 2 & 0 & - 12\end{array} \right]$ by the method of expansion by cofactors using column 3. Show your work.

A convention planning service has identified three suitable hotels for a convention. The quoted room rates are for single, double, triple, and quadruple occupancy. The current cost for each type of room at each hotel is represented by the matrix $A$ $A = \left[ \begin{array} { l l l } 85 & 91 & 98 \\95 & 103 & 110 \\103 & 109 & 120 \\113 & 121 & 129\end{array} \right]$ where the columns correspond to each hotel and the rows correspond to single, double, triple, and quadruple occupancy respectively. If room rates are guaranteed not to increase by more than 10% by the time of the convention, what is the maximum rate per room per hotel?

Given $A = \left[ \begin{array} { c c c } 0 & 3 & 6 \\- 9 & - 6 & 3 \\0 & 12 & 3\end{array} \right]$ and $B = \left[ \begin{array} { c c c } 9 & - 6 & 0 \\3 & - 3 & 6 \\9 & 3 & 3\end{array} \right]$ , find $| B A |$ .