91Ó°ÊÓ

Determine whether the product \(A B\) or \(B A\) is defined. If a product is defined, state its size ( number of rows and columns). Do not actually calculate any products. $$A=\left(\begin{array}{ccc} 0 & 5 & 9 \\ 0 & 5 & 3 \\ 0 & 0 & 0 \end{array}\right), \quad B=\left(\begin{array}{ccc} -8 & 7 & 1 \\ 0 & -3 & 4 \end{array}\right)$$

Determine whether the product \(A B\) or \(B A\) is defined. If a product is defined, state its size ( number of rows and columns). Do not actually calculate any products. $$A=\left(\begin{array}{rr} -4 & 15 \\ 3 & -7 \\ 2 & 10 \end{array}\right), \quad B=\left(\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right)$$

Show that AB is not equal to BA by computing both products. $$A=\left(\begin{array}{rrrr} -1 & 3 & 2 & -4 \\ 8 & 0 & 5 & 6 \\ -1 & 0 & 6 & 3 \\ 2 & 3 & -2 & 5 \end{array}\right), \quad B=\left(\begin{array}{rrrr} 4 & -2 & -2 & 0 \\ 1 & -7 & 4 & 1 \\ 0 & 5 & 7 & -2 \\ 1 & 4 & 0 & 5 \end{array}\right)$$

Use the elimination method to solve the system. $$\begin{aligned}&\frac{x+y}{4}-\frac{x-y}{3}=1\\\&\frac{x+y}{4}+\frac{x-y}{2}=9\end{aligned}$$

Exercises \(37-40,\) solve the system. [Note: The REF and RREF keys on some calculators produce an error message when there are more rows than columns in a matrix, in which case you will have to solve the system by some other means.] $$\begin{array}{r} x+y=3 \\ -x+2 y=3 \\ 5 x-y=3 \\ -7 x+5 y=3 \end{array}$$

Find the equilibrium quantity and the equilibrium price. In the supply and demand equations, \(p\) is price (in dollars) and \(x\) is quantity (in thousands). Supply: \(p=300-30 x\) Demand: \(p=80+25 x\)

A boat made a 4-mile trip upstream against a constant current in 15 minutes. The return trip at the same constant specd with the same current took 12 minutes. What is the speed of the boat and what is the speed of the current?

A plane flying into a headwind travels 2000 miles in 4 hours and 24 minutes. The return flight along the same= route with a tailwind takes 4 hours. Find the wind speed and the plane's speed (assuming that both are constant).

A candy company produces three types of gift boxes: \(A, B,\) and \(C .\) A box of variety \(A\) contains .6 pound of chocolates and .4 pound of mints. A box of variety \(B\) contains .3 pound of chocolates, .4 pound of mints, and .3 pound of caramels. A box of variety \(C\) contains .5 pound of chocolates, .3 pound of mints, and .2 pound of caramels. The company has 41,400 pounds of chocolates, 29,400 pounds of mints, and 16,200 pounds of caramels in stock. How many boxes of each variety should be made to use up all the stock?

Peanuts cost \(\$ 3\) per pound, almonds cost \(\$ 4\) per pound, and cashews costs \(\$ 8\) per pound. How many pounds of each should be used to produce 140 pounds of a mixture costing S6 per pound, in which there are twice as many peanuts as almonds?