Exam 6: Systems of Linear Equations and Matrices

The diagram shows the roads connecting four cities. The matrix A below represents the number of routes between each pair of cities without passing through another city. Calculate $\mathrm{A}^{2}$ . What information is given by the entry in row 3, column 2 of $\mathrm{A}^{2}$ ?

Use the Gauss-Jordan method to solve the system of equations. - $x+4 y+3 z=11$ $3 y+4 z=19$ $\mathrm{z}=4$

a system of linear equations containing more equations than variables is always inconsistent.

Determine whether the two matrices are inverses of each other by computing their product. - $\left[\begin{array}{rr}-2 & 4 \\ 4 & -4\end{array}\right],\left[\begin{array}{ll}\frac{1}{2} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{4}\end{array}\right]$

If $A+B=A$ and $A$ is a $3 \times 2$ matrix, what can you say about the matrix $B$ ?

Write a matrix to display the information. -The matrix shows the average number of wax and buff treatments each of 3 workers in a car wash can do in a day. Give the matrix that shows what each worker can do in 3 days.

Solve the matrix equation for $X$ .} - $\mathrm{A}=\left[\begin{array}{rr}3 & -1 \\ 3 & 0\end{array}\right], \mathrm{B}=\left[\begin{array}{rr}-3 & -9 \\ 0 & -3\end{array}\right], \mathrm{AX}=\mathrm{B}$

Find the inverse, if it exists, of the given matrix. - $\left[\begin{array}{rrr}2 & -1 & 0 \\ 3 & -2 & 0 \\ -2 & 3 & 1\end{array}\right]$

Write the word or phrase that best completes each statement or answers thequestion. -Suppose the following matrix represents the input-output matrix of a simplified economy. How many units of each commodity should be produced to satisfy a demand of 1400 units for each commodity?

Solve the problem. -During rush hours, substantial traffic congestion is encountered at the intersections shown in the figure. The arrows indicate one-way streets. As the figure shows, 500 cars per hour come down P Street to intersection A, and 200 cars per hour come down 5th Street to intersection A. x of these cars leave A on P Street and w cars leave A on 5th Street. The number of cars entering intersection A must equal the number leaving, so that $x+w=200+500$ or $x+w=700$ . By writing an equation representing the traffic entering and leaving each of the intersections A, B, C, and D, obtain a system of four equations. Solve the system using $\mathrm{w}$ as the parameter.

Find the production matrix for the input-output and demand matrices. - $\mathrm{A}=\left[\begin{array}{rr}\frac{1}{2} & \frac{2}{5} \\ \frac{1}{6} & \frac{2}{3}\end{array}\right], \mathrm{D}=\left[\begin{array}{r}6 \\ 10\end{array}\right]$

Perform the indicated operation. -Let $\mathrm{A}=\left[\begin{array}{rr}-3 & 5 \\ 0 & 2\end{array}\right]$ . Find $4 \mathrm{~A}$ .

Solve the matrix equation for $X$ .} - $\mathrm{A}=\left[\begin{array}{rr}-1 & 3 \\ 1 & 4\end{array}\right], \mathrm{B}=\left[\begin{array}{rrr}3 & -7 & 2 \\ 4 & -14 & 12\end{array}\right], \mathrm{AX}=\mathrm{B}$

Solve the problem. -Three different clothing stores order the following amounts of clothing by a certain designer: The unit prices of each product are given below for two suppliers: What matrix product displays the cost to each store of buying the clothes from each supplier? Display the two matrices which must be multiplied and their product.

The diagram shows the roads connecting four cities. How many ways are there to travel between cities $\mathrm{W}$ and $\mathrm{Z}$ by passing through exactly one city? (Hint: Write a matrix, A, to represent the number of routes between each pair of cities without passing through another city. Then calculate $\mathrm{A}^{2}$ ).

Solve the problem by writing and solving a suitable system of equations. -Factories A and B sent rice to stores 1 and 2. A sent 12 loads and B sent 24 . Store 1 used 19 loads and store 2 used 17. It cost $\$ 200$ to ship from A to 1, \$350 from A to 2, $\$ 300$ from B to 1, and $\$ 250$ from B to 2 . $\$ 8750$ was spent. How many loads went where?

Solve the problem by writing and solving a suitable system of equations. -A company makes 3 types of cable. Cable A requires 3 black, 3 white, and 2 red wires. Cable B requires 1 black, 2 white, and 1 red wires. Cable $C$ requires 2 black, 1 white, and 2 red wires. They used 100 black, 110 white and 80 red wires. How many of each cable were made?

Write an augmented matrix for the system of equations. - $2 x+4 y=22$ $4 y=4$

Write a matrix to display the information. -Carney and Dobler sell home and mortgage insurance. Their sales for the months of May and June are given in the matrices. Find the matrix that would give the change in sales from May to June.

Solve the system of two equations in two variables. - $-7 x-7 y=-56$ $-2 x+5 y=-16$