91Ó°ÊÓ

a. Give the order of each matrix. b. If \(A=\left[a_{i j}\right],\) identify \(a_{32}\) and \(a_{23},\) or explain why identification is not possible. $$ \left[\begin{array}{rrrr} {-4} & {1} & {3} & {-5} \\ {2} & {-1} & {\pi} & {0} \\ {1} & {0} & {-e} & {\frac{1}{5}} \end{array}\right] $$

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{rrr} {-2} & {1} & {-1} \\ {-5} & {2} & {-1} \\ {3} & {-1} & {1} \end{array}\right], \quad B=\left[\begin{array}{rrr} {1} & {0} & {1} \\ {2} & {1} & {3} \\ {-1} & {1} & {1} \end{array}\right] $$

Write a system of linear equations in three or four variables to solve. Then use matrices to solve the system. Three foods have the following nutritional content per ounce. $$ \begin{array}{lccc} {} & {} & {\text { Protein }} & {\text { Vitamin } \mathrm{C}} \\ & {\text { Calories }} & {\text { (in grams) }} & {\text { (in milligrams) }} \\\ \hline \text { Food } A & {40} & {5} & {30} \\ {\text { Food } B} & {200} & {2} & {10} \\ {\text { Food } C} & {400} & {4} & {300} \end{array} $$ If a meal consisting of the three foods allows exactly 660 calories, 25 grams of protein, and 425 milligrams of vitamin \(\mathrm{C},\) how many ounces of each kind of food should be used?

The table shows the daily production level and profit for a business. y (Daily Profit) $$ \begin{array}{ll} {x \text { (Number of Units }} & {30} & {50} & {100} \\ {\text { Produced Daily) }} \\ {y \text { (Daily Profit) }} & {\$ 5900} & {\$ 7500} & {\$ 4500} \end{array} $$ Use the quadratic function \(y=a x^{2}+b x+c\) to determine the number of units that should be produced each day for maximum profit. What is the maximum daily profit?

Solve: \(2 \cos ^{2} x+3 \sin x-3=0, \quad 0 \leq x<2 \pi\)