Chapter 12

Maximize the function \(f(x, y)=3 x+7 y\) in the region determined by the following constraints: 63 $$ \begin{aligned} 3 x+2 y & \leq 18 \\ 3 x+4 y & \geq 12 \\ x & \geq 0 \\ y & \geq 0 \end{aligned} $$

For Problems 21-36, use the technique discussed in this section to find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rrr} 1 & 4 & -2 \\ -3 & -11 & 1 \\ 2 & 7 & 3 \end{array}\right] $$

For Problems 27-40, use the method of matrix inverses to solve each system. $$ \left(\begin{array}{rl} 6 x-y & =-14 \\ 3 x+2 y & =-17 \end{array}\right) $$

For Problems \(27-30\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} -2 & 3 \\ 5 & 4 \end{array}\right] \quad B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right] \quad D=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]\\\ &I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned}\) $$ \text { Compute } A I \text { and } I A \text {. } $$

Maximize the function \(f(x, y)=1.5 x+2 y\) in the region determined by the following constraints: \(\quad 21\) $$ \begin{aligned} 3 x+2 y & \leq 36 \\ 3 x+10 y & \leq 60 \\ x & \geq 0 \\ y & \geq 0 \end{aligned} $$

For Problems 21-36, use the technique discussed in this section to find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rrr} 2 & 3 & -4 \\ 3 & -1 & -2 \\ 1 & -4 & 2 \end{array}\right] $$

For Problems 27-40, use the method of matrix inverses to solve each system. $$ \left(\begin{array}{rl} x-7 y & =7 \\ 6 x+5 y & =-5 \end{array}\right) $$

For Problems \(31-34\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} 2 & 4 \\ 5 & -3 \end{array}\right] \quad B=\left[\begin{array}{ll} -2 & 3 \\ -1 & 2 \end{array}\right] \\ &C=\left[\begin{array}{ll} 2 & 1 \\ 3 & 7 \end{array}\right] \end{aligned}\) $$ \text { Show that }(A B) C=A(B C) \text {. } $$

Maximize the function \(f(x, y)=40 x+55 y\) in the region determined by the following constraints: \(\quad 340\) $$ \begin{aligned} 2 x+y & \leq 10 \\ x+y & \leq 7 \\ 2 x+3 y & \leq 18 \\ x & \geq 0 \\ y & \geq 0 \end{aligned} $$

For Problems 27-40, use the method of matrix inverses to solve each system. $$ \left(\begin{array}{rl} x+9 y & =-5 \\ 4 x-7 y & =-20 \end{array}\right) $$

For Problems \(31-34\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} 2 & 4 \\ 5 & -3 \end{array}\right] \quad B=\left[\begin{array}{ll} -2 & 3 \\ -1 & 2 \end{array}\right] \\ &C=\left[\begin{array}{ll} 2 & 1 \\ 3 & 7 \end{array}\right] \end{aligned}\) $$ \text { Show that } A(B+C)=A B+A C $$

Maximize the function \(f(x, y)=0.08 x+0.09 y\) in the region determined by the following constraints: 660 $$ \begin{aligned} x+y & \leq 8000 \\ y & \leq \frac{1}{3} x \\ y & \geq 500 \\ x & \leq 7000 \\ x & \geq 0 \end{aligned} $$

For Problems 21-36, use the technique discussed in this section to find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rrr} 1 & 2 & 3 \\ -3 & -4 & 3 \\ 2 & 4 & -1 \end{array}\right] $$

For Problems 27-40, use the method of matrix inverses to solve each system. $$ \left(\begin{array}{l} 3 x-5 y=2 \\ 4 x-3 y=-1 \end{array}\right) $$

For Problems \(31-34\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} 2 & 4 \\ 5 & -3 \end{array}\right] \quad B=\left[\begin{array}{ll} -2 & 3 \\ -1 & 2 \end{array}\right] \\ &C=\left[\begin{array}{ll} 2 & 1 \\ 3 & 7 \end{array}\right] \end{aligned}\) $$ \text { Show that }(A+B) C=A C+B C \text {. } $$

Minimize the function \(f(x, y)=0.2 x+0.5 y\) in the region determined by the following constraints: 2 $$ \begin{aligned} 2 x+y & \geq 12 \\ 2 x+5 y & \geq 20 \\ x & \geq 0 \\ y & \geq 0 \end{aligned} $$

For Problems \(31-34\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} 2 & 4 \\ 5 & -3 \end{array}\right] \quad B=\left[\begin{array}{ll} -2 & 3 \\ -1 & 2 \end{array}\right] \\ &C=\left[\begin{array}{ll} 2 & 1 \\ 3 & 7 \end{array}\right] \end{aligned}\) $$ \text { Show that }(3+2) A=3 A+2 A \text {. } $$

Minimize the function \(f(x, y)=3 x+7 y\) in the region determined by the following constraints: 42 $$ \begin{aligned} x+y & \geq 9 \\ 6 x+11 y & \geq 84 \\ x & \geq 0 \\ y & \geq 0 \end{aligned} $$

For Problems 21-36, use the technique discussed in this section to find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{rrr} 2 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 10 \end{array}\right] $$

For Problems 27-40, use the method of matrix inverses to solve each system. $$ \left(\begin{array}{l} y=19-3 x \\ 9 x-5 y=1 \end{array}\right) $$

For Problems \(35-43\), use the following matrices. \( \begin{aligned} A &=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] & B &=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right] \\ C &=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] & O &=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \end{aligned} \) Show that \(A+B=B+A\).

Maximize the function \(f(x, y)=9 x+2 y\) in the region determined by the following constraints: 98 $$ \begin{aligned} 5 y-4 x & \leq 20 \\ 4 x+5 y & \leq 60 \\ x & \geq 0 \\ x & \leq 10 \\ y & \geq 0 \end{aligned} $$

For Problems 27-40, use the method of matrix inverses to solve each system. $$ \left(\begin{array}{l} 4 x+3 y=31 \\ x=5 y+2 \end{array}\right) $$

For Problems \(35-43\), use the following matrices. \( \begin{aligned} A &=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] & B &=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right] \\ C &=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] & O &=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \end{aligned} \) Show that \((A+B)+C=A+(B+C)\).

Maximize the function \(f(x, y)=3 x+4 y\) in the region determined by the following constraints: 42 $$ \begin{aligned} 2 y-x & \leq 6 \\ x+y & \leq 12 \\ x & \geq 2 \\ x & \leq 8 \\ y & \geq 0 \end{aligned} $$

For Problems 27-40, use the method of matrix inverses to solve each system. $$ \left(\begin{array}{c} 3 x+2 y=0 \\ 30 x-18 y=-19 \end{array}\right) $$

For Problems \(35-43\), use the following matrices. \( \begin{aligned} A &=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] & B &=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right] \\ C &=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] & O &=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \end{aligned} \) Show that \((A+B)+C=A+(B+C)\).

For Problems 27-40, use the method of matrix inverses to solve each system. $$ \left(\begin{array}{l} 12 x+30 y=23 \\ 12 x-24 y=-13 \end{array}\right) $$

For Problems \(35-43\), use the following matrices. \( \begin{aligned} A &=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] & B &=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right] \\ C &=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] & O &=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \end{aligned} \) Show that \(k(A+B)=k A+k B\) for any real number \(k\).

A manufacturer of golf clubs makes a profit of \(\$ 50\) per set on a model A set and \(\$ 45\) per set on a model \(B\) set. Daily production of the model A clubs is between 30 and 50 sets, inclusive, and that of the model B clubs is between 10 and 20 sets, inclusive. The total daily production is not to exceed 50 sets. How many sets of each model should be manufactured per day to maximize the profit?

For Problems 27-40, use the method of matrix inverses to solve each system. $$ \left(\begin{array}{l} \frac{1}{3} x+\frac{3}{4} y=12 \\ \frac{2}{3} x+\frac{1}{5} y=-2 \end{array}\right) $$

For Problems \(35-43\), use the following matrices. \( \begin{aligned} A &=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] & B &=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right] \\ C &=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] & O &=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \end{aligned} \) Show that \((k+l) A=k A+l A\) for any real numbers \(k\) and \(l\).

A company makes two types of calculators. Type A sells for \(\$ 12\), and type B sells for \(\$ 10\). It costs the company \(\$ 9\) to produce one type A calculator and \(\$ 8\) to produce one type B calculator. In one month, the company is equipped to produce between 200 and 300 , inclusive, of the type A calculator and between 100 and 250 , inclusive, of the type B calculator, but not more than 300 altogether. How many calculators of each type should be produced per month to maximize the difference between the total selling price and the total cost of production?

For Problems 27-40, use the method of matrix inverses to solve each system. $$ \left(\begin{array}{l} \frac{3}{2} x+\frac{1}{6} y=11 \\ \frac{2}{3} x-\frac{1}{4} y=1 \end{array}\right) $$

For Problems \(35-43\), use the following matrices. \( \begin{aligned} A &=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] & B &=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right] \\ C &=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] & O &=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \end{aligned} \) Show that \((k l) A=k(l A)\) for any real numbers \(k\) and \(l\).

A manufacturer of small copiers makes a profit of \(\$ 200\) on a deluxe model and \(\$ 250\) on a standard model. The company wants to produce at least 50 deluxe models per week and at least 75 standard models per week. However, the weekly production is not to exceed 150 copiers. How many copiers of each kind should be produced in order to maximize the profit?

Describe how to solve the system \(\left(\begin{array}{rl}x-2 y & =-10 \\ 3 x+5 y & =14\end{array}\right)\) using each of the following techniques. (a) substitution method (b) elimination-by-addition method (c) reduced echelon form of the augmented matrix (d) determinants (e) the method of matrix inverses

Products \(\mathrm{A}\) and \(\mathrm{B}\) are produced by a company according to the following production information. (a) To produce one unit of product A requires 1 hour of working time on machine I, 2 hours on machine II, and 1 hour on machine III. (b) To produce one unit of product B requires 1 hour of working time on machine I, 1 hour on machine II, and 3 hours on machine III. (c) Machine I is available for no more than 40 hours per week, machine II for no more than 40 hours per week, and machine III for no more than 60 hours per week. (d) Product \(A\) can be sold at a profit of \(\$ 2.75\) per unit and product B at a profit of \(\$ 3.50\) per unit. How many units each of product \(A\) and product \(B\) should be produced per week to maximize profit?

Use your calculator to find the multiplicative inverse (if one exists) of each of the following matrices. Be sure to check your answers by showing that \(A^{-1} A=I\). (a) \(\left[\begin{array}{ll}7 & 6 \\ 8 & 7\end{array}\right]\) (b) \(\left[\begin{array}{ll}-12 & 5 \\ -19 & 8\end{array}\right]\) (c) \(\left[\begin{array}{rr}-7 & 9 \\ 6 & -8\end{array}\right]\) (d) \(\left[\begin{array}{rr}-6 & -11 \\ -4 & -8\end{array}\right]\) (e) \(\left[\begin{array}{rr}13 & 12 \\ 4 & 4\end{array}\right]\) (f) \(\left[\begin{array}{rr}15 & -8 \\ -9 & 5\end{array}\right]\) (g) \(\left[\begin{array}{ll}9 & 36 \\ 3 & 12\end{array}\right]\) (h) \(\left[\begin{array}{ll}1.2 & 1.5 \\ 7.6 & 4.5\end{array}\right]\)

For Problems \(35-43\), use the following matrices. \( \begin{aligned} A &=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] & B &=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right] \\ C &=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] & O &=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \end{aligned} \) Show that \(A(B+C)=A B+A C\).

Use your calculator to find the multiplicative inverse of \(\left[\begin{array}{ll}\frac{1}{2} & \frac{2}{5} \\ \frac{3}{4} & \frac{1}{4}\end{array}\right]\) What difficulty did you encounter?

For Problems \(35-43\), use the following matrices. \( \begin{aligned} A &=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] & B &=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right] \\ C &=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] & O &=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \end{aligned} \) Show that \((A+B) C=A C+B C\).

Describe in your own words the process of solving a system of inequalities.

Use your calculator and the method of matrix inverses to solve each of the following systems. Be sure to check your solutions. (a) \(\left(\begin{array}{c}5 x+7 y=82 \\ 7 x+10 y=116\end{array}\right)\) (b) \(\left(\begin{array}{rl}9 x-8 y & =-150 \\ -10 x+9 y & =168\end{array}\right)\) (c) \(\left(\begin{array}{rl}15 x-8 y & =-15 \\ -9 x+5 y & =12\end{array}\right)\) (d) \(\left(\begin{array}{l}1.2 x+1.5 y=5.85 \\ 7.6 x+4.5 y=19.55\end{array}\right)\) (e) \(\left(\begin{array}{c}12 x-7 y=-34.5 \\ 8 x+9 y=79.5\end{array}\right)\) (f) \(\left(\begin{array}{l}\frac{3 x}{2}+\frac{y}{6}=11 \\ \frac{2 x}{3}-\frac{y}{4}=1\end{array}\right)\) (g) \(\left(\begin{array}{l}114 x+129 y=2832 \\ 127 x+214 y=4139\end{array}\right)\) (h) \(\left(\begin{array}{l}\frac{x}{2}+\frac{2 y}{5}=14 \\ \frac{3 x}{4}+\frac{y}{4}=14\end{array}\right)\)

How would you show that addition of \(2 \times 2\) matrices is a commutative operation?

What is linear programming? Write a paragraph or two answering this question in a way that elementary algebra students could understand.

How would you show that subtraction of \(2 \times 2\) matrices is not a commutative operation?

Your friend says that because multiplication of real numbers is a commutative operation, it seems reasonable that multiplication of matrices should also be a commutative operation. How would you react to that statement?

How would you describe row-by-column multiplication of matrices?

If \(A=\left[\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right]\), calculate \(A^{2}\) and \(A^{3}\), where \(A^{2}\) means \(A A\), and \(A^{3}\) means \(A A A . \)