Chapter 6

The matrix $$ P=\left[\begin{array}{lll} 0.6 & 0.1 & 0.1 \\ 0.2 & 0.7 & 0.1 \\ 0.2 & 0.2 & 0.8 \end{array}\right] $$ is called a stochastic matrix. Each entry \(p_{i j}(i \neq j)\) represents the proportion of the voting population that changes from Party \(i\) to Party \(j\), and \(p_{i i}\) represents the proportion that remains loyal to the party from one election to the next. Use a graphing utility to find \(P^{2}\). (This matrix gives the transition probabilities from the first election to the third.)

Evaluate the determinant of the matrix. Do not use a graphing utility. $$ \left[\begin{array}{rrrrr} -3 & 0 & 0 & 0 & 0 \\ 4 & 1 & 0 & 0 & 0 \\ 7 & -8 & 7 & 0 & 0 \\ 6 & 4 & 0 & -2 & 0 \\ 1 & 5 & 1 & -10 & 6 \end{array}\right] $$

You invest in AAA-rated bonds, A-rated bonds, and B-rated bonds. Your average yield is \(9 \%\) on AAA bonds, \(7 \%\) on \(A\) bonds, and \(8 \%\) on B bonds. You invest twice as much in \(\mathrm{B}\) bonds as in \(\mathrm{A}\) bonds. The desired system of linear equations (where \(x, y\), and \(z\) represent the amounts invested in AAA, A, and B bonds, respectively) is as follows. \(\left\\{\begin{aligned} x+y+z &=\text { (total investment) } \\ 0.09 x+0.07 y+0.08 z &=\text { (annual return) } \\ 2 y-\quad z &=0 \end{aligned}\right.\) Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond for the given total investment and annual return. $$ \text { Total investment }=\$ 50,000 ; \text { annual return }=\$ 4180 $$

In Exercises \(71-74\), find the determinant of the matrix. Tell which method you used. $$ \left[\begin{array}{rrr} 2 & 1 & 3 \\ 7 & 3 & -2 \\ 4 & 1 & 1 \end{array}\right] $$

You invest in AAA-rated bonds, A-rated bonds, and B-rated bonds. Your average yield is \(9 \%\) on AAA bonds, \(7 \%\) on \(A\) bonds, and \(8 \%\) on B bonds. You invest twice as much in \(\mathrm{B}\) bonds as in \(\mathrm{A}\) bonds. The desired system of linear equations (where \(x, y\), and \(z\) represent the amounts invested in AAA, A, and B bonds, respectively) is as follows. \(\left\\{\begin{aligned} x+y+z &=\text { (total investment) } \\ 0.09 x+0.07 y+0.08 z &=\text { (annual return) } \\ 2 y-\quad z &=0 \end{aligned}\right.\) Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond for the given total investment and annual return. $$ \text { Tutal investuent }-\$ 36,000 \text { ; aumual return }-\$ 3040 $$

Find a matrix \(B\) such that \(A B\) is the identity matrix. Is there more than one correct result? $$ A=\left[\begin{array}{ll} 1 & 3 \\ 1 & 2 \end{array}\right] $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} 3 x+3 y+12 z=& 6 \\ x+y+4 z=& 2 \\ 2 x+5 y+20 z=& 10 \\ -x+2 y+8 z=& 4 \end{aligned}\right. $$

Find (a) \(|A|\), (b) \(|B|\), (c) \(A B\), and (d) \(|A B|\). $$ \left[\begin{array}{rrr} 6 & -5 & 2 \\ 0 & 5 & -3 \\ 0 & 0 & 2 \end{array}\right] $$

In Exercises \(71-74\), find the determinant of the matrix. Tell which method you used. $$ \left[\begin{array}{rrr} 6 & -5 & 2 \\ 0 & 5 & -3 \\ 0 & 0 & 2 \end{array}\right] $$

You invest in AAA-rated bonds, A-rated bonds, and B-rated bonds. Your average yield is \(9 \%\) on AAA bonds, \(7 \%\) on \(A\) bonds, and \(8 \%\) on B bonds. You invest twice as much in \(\mathrm{B}\) bonds as in \(\mathrm{A}\) bonds. The desired system of linear equations (where \(x, y\), and \(z\) represent the amounts invested in AAA, A, and B bonds, respectively) is as follows. \(\left\\{\begin{aligned} x+y+z &=\text { (total investment) } \\ 0.09 x+0.07 y+0.08 z &=\text { (annual return) } \\ 2 y-\quad z &=0 \end{aligned}\right.\) Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond for the given total investment and annual return. $$ \text { Total investment }=\$ 45,000 ; \text { annual return }=\$ 3770 $$

Find a matrix \(B\) such that \(A B\) is the identity matrix. Is there more than one correct result? $$ A=\left[\begin{array}{ll} 2 & 1 \\ 5 & 2 \end{array}\right] $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} 2 x+10 y+2 z &=6 \\ x+5 y+2 z &=6 \\ x+5 y+z &=3 \\ -3 x+15 y-3 z &=-9 \end{aligned}\right. $$

In Exercises \(71-74\), find the determinant of the matrix. Tell which method you used. $$ \left[\begin{array}{rrr} 3 & 0 & 0 \\ 4 & -2 & 0 \\ 5 & 4 & 3 \end{array}\right] $$

Circuit Analysis In Exercises 73 and 74, consider the circuit shown in the figure. The currents \(I_{1}, I_{2}\), and \(I_{3}\), in amperes, are the solution of the system of linear equations \(\left\\{\begin{aligned} 2 I_{1} \quad+4 I_{3}=& E_{1} \\ I_{2}+4 I_{3}=& E_{2} \\ I_{1}+I_{2}-I_{3}=& 0 \end{aligned}\right.\) where \(E_{1}\) and \(E_{2}\) are voltages. Use the inverse of the coefficient matrix of this system to find the unknown currents for the given voltages. \(E_{1}=28\) volts, \(E_{2}=21\) volts

If \(a, b\), and \(c\) are real numbers such that \(c \neq 0\) and \(a c=b c\), then \(a=b\). However, if \(A, B\), and \(C\) are matrices such that \(A C=B C\), then \(A\) is not necessarily equal to \(B\). Illustrate this using the following matrices. \(A=\left[\begin{array}{rrr}1 & 2 & 3 \\ 0 & 5 & 4 \\ 3 & -2 & 1\end{array}\right], B=\left[\begin{array}{rrr}4 & -6 & 3 \\ 5 & 4 & 4 \\ -1 & 0 & 1\end{array}\right]\), and \(C=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 4 & -2 & 3\end{array}\right]\)

In Exercises \(71-74\), find the determinant of the matrix. Tell which method you used. $$ \left[\begin{array}{rrr} 3 & 2 & -4 \\ -1 & 5 & -3 \\ 0 & 1 & 0 \end{array}\right] $$

Circuit Analysis In Exercises 73 and 74, consider the circuit shown in the figure. The currents \(I_{1}, I_{2}\), and \(I_{3}\), in amperes, are the solution of the system of linear equations \(\left\\{\begin{aligned} 2 I_{1} \quad+4 I_{3}=& E_{1} \\ I_{2}+4 I_{3}=& E_{2} \\ I_{1}+I_{2}-I_{3}=& 0 \end{aligned}\right.\) where \(E_{1}\) and \(E_{2}\) are voltages. Use the inverse of the coefficient matrix of this system to find the unknown currents for the given voltages. \(E_{1}=24\) volts, \(E_{2}=23\) volts

If \(a\) and \(b\) are real numbers such that \(a b=0\), then \(a=0\) or \(b=0\). However, if \(A\) and \(B\) are matrices such that \(A B=O\), then it is not necessarily true that \(A=O\) or \(B=O\). Illustrate this using the following matrices. \(A=\left[\begin{array}{ll}3 & 3 \\ 4 & 4\end{array}\right]\) and \(B=\left[\begin{array}{rr}1 & -1 \\ -1 & 1\end{array}\right]\) Find another example of two nonzero matrices whose product is the zero matrix.

Find \((a)|A|\), (b) \(|B|\), (c) \(A B\), and \((d)\) \(|A B|\). $$ A=\left[\begin{array}{rr} -1 & 0 \\ 0 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 0 \\ 0 & -1 \end{array}\right] $$

Consider a company that specializes in potting soil. Each bag of potting soil for seedlings requires 2 units of sand, 1 unit of loam, and 1 unit of peat moss. Each bag of potting soil for general potting requires 1 unit of sand, 2 units of loam, and 1 unit of peat moss. Each bag of potting soil for hardwood plants requires 2 units of sand, 2 units of loam, and 2 units of peat moss. Find the numbers of bags of the three types of potting soil that the company can produce with the given amounts of raw materials. 500 units of sand 500 units of loam 400 units of peat moss

Determine whether the statement is true or false. Justify your answer. $$ \left[\begin{array}{rr} 3 & 2 \\ 1 & -4 \end{array}\right]\left[\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\left[\begin{array}{rr} 3 & 2 \\ 1 & -4 \end{array}\right] $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{r} x+2 y=0 \\ -x-y=0 \end{array}\right. $$

Find \((a)|A|\), (b) \(|B|\), (c) \(A B\), and \((d)\) \(|A B|\). $$ A=\left[\begin{array}{rr} -2 & 1 \\ 4 & -2 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array}\right] $$

Consider a company that specializes in potting soil. Each bag of potting soil for seedlings requires 2 units of sand, 1 unit of loam, and 1 unit of peat moss. Each bag of potting soil for general potting requires 1 unit of sand, 2 units of loam, and 1 unit of peat moss. Each bag of potting soil for hardwood plants requires 2 units of sand, 2 units of loam, and 2 units of peat moss. Find the numbers of bags of the three types of potting soil that the company can produce with the given amounts of raw materials. 500 units of sand 750 units of loam 450 units of peat moss

Determine whether the statement is true or false. Justify your answer. $$ \left[\begin{array}{rr} -6 & -2 \\ 2 & -6 \end{array}\right]\left[\begin{array}{rr} 4 & 0 \\ 0 & -1 \end{array}\right]=\left[\begin{array}{rr} 4 & 0 \\ 0 & -1 \end{array}\right]\left[\begin{array}{rr} -6 & -2 \\ 2 & -6 \end{array}\right] $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{r} x+2 y=0 \\ 2 x+4 y=0 \end{array}\right. $$

Find \((a)|A|\), (b) \(|B|\), (c) \(A B\), and \((d)\) \(|A B|\). $$ A=\left[\begin{array}{rr} 4 & 0 \\ 3 & -2 \end{array}\right], \quad B=\left[\begin{array}{ll} -1 & 1 \\ -2 & 2 \end{array}\right] $$

Consider a company that specializes in potting soil. Each bag of potting soil for seedlings requires 2 units of sand, 1 unit of loam, and 1 unit of peat moss. Each bag of potting soil for general potting requires 1 unit of sand, 2 units of loam, and 1 unit of peat moss. Each bag of potting soil for hardwood plants requires 2 units of sand, 2 units of loam, and 2 units of peat moss. Find the numbers of bags of the three types of potting soil that the company can produce with the given amounts of raw materials. 350 units of sand 445 units of loam 345 units of peat moss

Two competing companies offer cable television to a city with 100,000 households. Gold Cable Company has 25,000 subscribers and Galaxy Cable Company has 30,000 subscribers. (The other 45,000 households do not subscribe.) The percent changes in cable subscriptions each year are shown in the matrix below. $$ \left[\begin{array}{lll} 0.70 & 0.15 & 0.15 \\ 0.20 & 0.80 & 0.15 \\ 0.10 & 0.05 & 0.70 \end{array}\right] $$ (a) Find the number of subscribers each company will have in one year using matrix multiplication. Explain how you obtained your answer. (b) Find the number of subscribers each company will have in two years using matrix multiplication. Explain how you obtained your answer. (c) Find the number of subscribers each company will have in three years using matrix multiplication. Explain how you obtained your answer. (d) What is happening to the number of subscribers to each company? What is happening to the number of nonsubscribers?

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{r} x+y+z=0 \\ 2 x+3 y+z=0 \\ 3 x+5 y+z=0 \end{array}\right. $$

Find \((a)|A|\), (b) \(|B|\), (c) \(A B\), and \((d)\) \(|A B|\). $$ A=\left[\begin{array}{rr} 5 & 4 \\ 3 & -1 \end{array}\right], \quad B=\left[\begin{array}{rr} 0 & 6 \\ 1 & -2 \end{array}\right] $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{r} x-2 y+z+3 w=0 \\ x-y+w=0 \\ y-z+2 w=0 \end{array}\right. $$

Find \((a)|A|\), (b) \(|B|\), (c) \(A B\), and \((d)\) \(|A B|\). $$ A=\left[\begin{array}{rrr} 0 & 1 & 2 \\ -3 & -2 & 1 \\ 0 & 4 & 1 \end{array}\right], \quad B=\left[\begin{array}{rrr} 3 & -2 & 0 \\ 1 & -1 & 2 \\ 3 & 1 & 1 \end{array}\right] $$

The total values \(y\) (in billions of dollars) of child support collections from 1998 to 2005 are shown in the figure. The least squares regression parabola \(y=a t^{2}+b t+c\) for these data is found by solving the system $$ \left\\{\begin{array}{r} 8 c+92 b+1100 a=153.3 \\ 92 c+1100 b+13,616 a=1813.9 \\ 1100 c+13,616 b+173,636 a=22,236.7 \end{array}\right. $$ (a) Use a graphing utility to find an inverse matrix to solve this system, and find the equation of the least squares regression parabola. (b) Use the result from part (a) to estimate the value of child support collections in \(2007 .\) (c) An analyst predicted that the value of child support collections in 2007 would be \(\$ 24.0\) billion. How does this value compare with your estimate in part (b)? Do both estimates seem reasonable?

Determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices. (a) \(\left\\{\begin{aligned} x-2 y+z &=-6 \\ y-5 z &=16 \\ z &=-3 \end{aligned}\right.\) (b) \(\left\\{\begin{aligned} x+y-2 z &=6 \\ y+3 z &=-8 \\ z &=-3 \end{aligned}\right.\)

Find \((a)|A|\), (b) \(|B|\), (c) \(A B\), and \((d)\) \(|A B|\). $$ A=\left[\begin{array}{rrr} 3 & 2 & 0 \\ -1 & -3 & 4 \\ -2 & 0 & 1 \end{array}\right], \quad B=\left[\begin{array}{rrr} -3 & 0 & 1 \\ 0 & 2 & -1 \\ -2 & -1 & 1 \end{array}\right] $$

The total annual profits \(y\) (in thousands of dollars) for an Alaskan fishing captain from 2000 to 2008 are shown in the figure. The least squares regression parabola \(y=a t^{2}+b t+c\) for these data is found by solving the system \(\left\\{\begin{aligned} 9 c+36 b+204 a &=1152 \\ 36 c+204 b+1296 a &=4399 . \\ 204 c+1296 b+8772 a &=24,597 \end{aligned}\right.\) Let \(t\) represent the year, with \(t=0\) corresponding to 2000 . (a) Use a graphing utility to find an inverse matrix to solve this system, and find the equation of the least squares regression parabola. (b) Use the result from part (a) to predict the captain's profit in 2010 . (c) Due to increased competition, the captain projects profits of \(\$ 115,000\) in \(2010 .\) How does this value compare with your prediction in part (b)?

Determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices. (a) \(\left\\{\begin{array}{rr}x-3 y+4 z= & -11 \\ y-z= & -4 \\ z= & 2\end{array}\right.\) (b) \(\left\\{\begin{array}{rr}x+4 y & =-11 \\ y+3 z & =4 \\ z & =2\end{array}\right.\)

Find \((a)|A|\), (b) \(|B|\), (c) \(A B\), and \((d)\) \(|A B|\). $$ A=\left[\begin{array}{rrr} -1 & 2 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array}\right] $$

Use the following matrices. $$A=\left[\begin{array}{rr}4 & 3 \\ -2 & 1\end{array}\right], B=\left[\begin{array}{rr}1 & -2 \\ 3 & 4\end{array}\right], C=\left[\begin{array}{rr}13 & 4 \\ 1 & 8\end{array}\right]$$ Find \(A B\) and \(B A\). What do you observe about the two products?

Determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices. (a) \(\left\\{\begin{aligned} x-4 y+5 z &=27 \\ y-7 z &=-54 \\ z &=8 \end{aligned}\right.\) (b) \(\left\\{\begin{aligned} x-6 y+z &=15 \\ y+5 z &=42 \\ z &=8 \end{aligned}\right.\)

Find \((a)|A|\), (b) \(|B|\), (c) \(A B\), and \((d)\) \(|A B|\). $$ A=\left[\begin{array}{rrr} 2 & 0 & 1 \\ 1 & -1 & 2 \\ 3 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & -1 & 4 \\ 0 & 1 & 3 \\ 3 & -2 & 1 \end{array}\right] $$

Use the following matrices. $$A=\left[\begin{array}{rr}4 & 3 \\ -2 & 1\end{array}\right], B=\left[\begin{array}{rr}1 & -2 \\ 3 & 4\end{array}\right], C=\left[\begin{array}{rr}13 & 4 \\ 1 & 8\end{array}\right]$$ Find \(C^{-1}, A^{-1} \cdot B^{-1}\), and \(B^{-1} \cdot A^{-1} .\) What do you observe about the three resulting matrices?

Determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices. (a) \(\left\\{\begin{aligned} x+3 y-z &=19 \\ y+6 z &=-18 \\ z &=-4 \end{aligned}\right.\) (b) \(\left\\{\begin{aligned} x-y+3 z &=-15 \\ y-2 z &=14 \\ z &=-4 \end{aligned}\right.\)

In Exercises 83 and 84 , find a value of \(k\) that makes the matrix invertible and then find a value of \(k\) that makes the matrix singular. (There are many correct answers.) $$ \left[\begin{array}{rr} 4 & 3 \\ -2 & k \end{array}\right] $$

A city zoo borrowed $$\$ 2,000,000$$ at simple annual interest to construct a breeding facility. Some of the money was borrowed at \(8 \%\), some at \(9 \%\), and some at \(12 \%\). Use a system of equations to determine how much was borrowed at each rate if the total annual interest was $$\$ 186,000$$ and the amount borrowed at \(8 \%\) was twice the amount borrowed at \(12 \%\). Solve the system using matrices.

In Exercises 83 and 84 , find a value of \(k\) that makes the matrix invertible and then find a value of \(k\) that makes the matrix singular. (There are many correct answers.) $$ \left[\begin{array}{cc} 2 k+1 & 3 \\ -7 & 1 \end{array}\right] $$

A natural history museum borrowed $$\$ 2,000,000$$ at simple annual interest to purchase new exhibits. Some of the money was borrowed at \(7 \%\), some at \(8.5 \%\), and some at \(9.5 \%\). Use a system of equations to determine how much was borrowed at each rate if the total annual interest was $$\$ 169,750$$ and the amount borrowed at \(8.5 \%\) was four times the amount borrowed at \(9.5 \%\). Solve the system using matrices.

In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. There exists a matrix \(A\) such that \(A=A^{-1}\).

You and a friend solve the following system of equations independently. \(\left\\{\begin{array}{rr}2 x-4 y-3 z= & 3 \\ x+3 y+z= & -1 \\ 5 x+y-2 z= & 2\end{array}\right.\) You write your solution set as \((a,-a, 2 a-1)\) where \(a\) is any real number. Your friend's solution set is \(\left(\frac{1}{2} b+\frac{1}{2},-\frac{1}{2} b-\frac{1}{2}, b\right)\) where \(b\) is any real number. Are you both correct? Explain. If you let \(a=3\), what value of \(b\) must be selected so that you both have the same ordered triple?