Exam 2: Systems of Linear Equations and Matrices

Solve the problem. -A basketball fieldhouse seats 15,000. Courtside seats cost $10, endzone seats cost $6, and balcony seats cost $4. The total revenue for a sellout is $82,000. If half the courtside seats, half the balcony Seats, and all the endzone seats are sold; then the total revenue is $47,000. How many of each type Of seat are there?

-ABC is a 2 × 4 matrix.

Find the matrix product, if possible. - $\left[\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right]\left[\begin{array}{ccc}5 & 2 & -2 \\ 2 & -2 & 2\end{array}\right]$

- AB is a 2 × 2 matrix.

Perform the indicated operation, where possible. - $\left[\begin{array}{r}3 \\-1 \\-4\end{array}\right]+\left[\begin{array}{r}-5 \\2 \\8\end{array}\right]$

Solve the problem. -Carole's car averages 15.3 miles per gallon in city driving and 24.5 miles per gallon in highway driving. If she drove a total of 382.7 miles on 19 gallons of gas, then how many of the gallons were Used for city driving?

Solve the system of equations by using the inverse of the coefficient matrix if it exists and by the echelon method if the inverse doesn't exist. - 3x+y=15 2x+4y=0

Use the Gauss-Jordan method to solve the system of equations. - $5 x-3 y=6$ $25 x-15 y=9$

For the following systems of equations in echelon form, tell how many solutions there are in nonnegative integers. - x+3y+4z=80 4y+5z=40

Solve the problem. -The matrices give points and rebounds for five starting players in two games. Find the matrix that gives the totals.

Use the echelon method to solve the system. - -=-18 +=-9

Solve the problem. -For their class play, Ron sold student tickets for $4.00 each and Kathy sold adult tickets for $6.50 each. If their total revenue for 29 tickets was $141.00, then how many tickets did Ron sell?

Solve the problem. -Carney and Dobler sell auto and hazard insurance. Their sales, in dollars, for the months of July and August are given in the following matrices. Auto Hazard July: $\left[\begin{array}{ll}22,000 & 45,000 \\ 19,000 & 27,000\end{array}\right]$ Carney August: $\left[\begin{array}{ll}25,000 & 44,000 \\ 14,000 & 21,000\end{array}\right]$ Carney Find a matrix that gives the increase (decrease) in sales by each salesman for each type of insurance from July to August.

Use a graphing calculator to solve the system of equations. Round your solution to one decimal place. - $2.6 x-0.7 y+5.3 z-3.4 w=12.9$ $6.5 \mathrm{y} \quad+0.7 \mathrm{w}=-1.8$ $4.0 z-2.2 w=1.5$ $4.9 x+0.6 w=3.2$

Decide whether the matrices are inverses of each other. (Check to see if their product is the identity matrix I.) - $\left[\begin{array}{ll}-5 & 1 \\ -7 & 1\end{array}\right]$ and $\left[\begin{array}{cc}\frac{1}{2} & -\frac{1}{2} \\ \frac{7}{2} & -\frac{5}{2}\end{array}\right]$

Provide an appropriate response. -What are the elements of the third row of the following matrix? 42) $\left[\begin{array}{rrrr}-2 & 1 & 3 & 4 \\-3 & -1 & 4 & 7 \\1 & -5 & 2 & 6\end{array}\right]$

Write a matrix to display the information. -A bakery sells three types of cakes. Cake I requires 2 cups of flour, 2 cups of sugar, and 2 eggs. Cake II requires 4 cups of flour, 1 cup of sugar, and 1 egg. Cake III requires 2 cups of flour, 2 cups of Sugar, and 3 eggs. Make a 3 × 3 matrix showing the required ingredients for each cake. Assign the Cakes to the rows and the ingredients to the columns.

Solve the problem. -A chemistry department wants to make 3 L of a 17.5% basic solution by mixing a 20% solution with a 15% solution. How many liters of each type of basic solution should be used to produce the 17.5% Solution?

Use a graphing calculator to find the matrix product and/or sum. -Find $\mathrm{AB}$ . $A=\left[\begin{array}{rrrr}18 & 14 & 18 & 18 \\30 & 5 & 7 & 2 \\11 & 3 & 2 & 1 \\5 & 32 & 5 & 7\end{array}\right] B=\left[\begin{array}{rrr}18 & 5 & 3 \\18 & 8 & 12 \\7 & 4 & 2 \\21 & 15 & 13\end{array}\right]$

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