Chapter 6

An augmented matrix that represents a system of linear equations (in variables \(x, y\), and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$ \left[\begin{array}{rrrrr} 1 & 0 & 2 & \vdots & 9 \\ 0 & 1 & 5 & \vdots & -3 \\ 0 & 0 & 0 & \vdots & 0 \end{array}\right] $$

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$ \left|\begin{array}{rrr} 7 & 0 & -14 \\ -2 & 5 & 4 \\ -6 & 2 & 12 \end{array}\right| $$

Inventory Levels A company sells five different models of computers through three retail outlets. The inventories of the five models at the three outlets are given by the matrix \(S\). $$ S=\left[\begin{array}{lllll} 3 & 2 & 2 & 3 & 0 \\ 0 & 2 & 3 & 4 & 3 \\ 4 & 2 & 1 & 3 & 2 \end{array}\right] $$ The wholesale and retail prices for each model are given by the matrix \(T\). $$ T=\left[\begin{array}{rl} \$ 900 & \$ 1200 \\ \$ 1200 & \$ 1450 \\ \$ 1400 & \$ 1650 \\ \$ 2650 & \$ 3250 \\ \$ 3050 & \$ 3375 \end{array}\right] $$ (a) What is the total retail price of the inventory at Outlet \(1 ?\) (b) What is the total wholesale price of the inventory at Outlet 3 ? (c) Compute the product \(S T\) and interpret the result in the context of the problem.

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$\left\\{\begin{array}{r}x+2 y=7 \\ 2 x+y=8\end{array}\right.$$

Your biology professor gives you the encoded message below. \(-204,47,-231,53,-265,61,-223,51,-9,2,-117, 28,-117,26,-166,37,-265,61,-145,34,-112,25,-76,19\) Let \(A^{-1}=\left[\begin{array}{rr}w & x \\ y & z\end{array}\right]\). You know that \(\left[\begin{array}{ll}-204 & 47\end{array}\right] A^{-1}=[1516]\) and that \(\left[\begin{array}{ll}-231 & 53\end{array}\right] A^{-1}=[15 \quad 19]\), where \(A^{-1}\) is the inverse of the encoding matrix \(A\). Explain how you can find the values of \(w, x, y\), and \(z\). Decode the message.

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$ \left|\begin{array}{rrr} 3 & 0 & 0 \\ -2 & 5 & 0 \\ 12 & 5 & 7 \end{array}\right| $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{l} 2 x+6 y=16 \\ 2 x+3 y=7 \end{array}\right. $$

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$ \left|\begin{array}{rrrr} 1 & -1 & 8 & 4 \\ 2 & 6 & 0 & -4 \\ 2 & 0 & 2 & 6 \\ 0 & 2 & 8 & 0 \end{array}\right| $$

Use an inverse matrix to solve (if possible) the system of linear equations. $$ \left\\{\begin{array}{l} 3 x+4 y=-2 \\ 5 x+3 y=4 \end{array}\right. $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{rr} -3 x+5 y= & -22 \\ 3 x+4 y= & 4 \\ 4 x-8 y= & 32 \end{array}\right. $$

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$ \left|\begin{array}{rrrr} 0 & -3 & 8 & 3 \\ 8 & 1 & -1 & 6 \\ -4 & 6 & 0 & 9 \\ -7 & 0 & 0 & 14 \end{array}\right| $$

Use an inverse matrix to solve (if possible) the system of linear equations. $$ \left\\{\begin{array}{l} 18 x+12 y=13 \\ 30 x+24 y=23 \end{array}\right. $$

A fruit grower raises apples and peaches, which are shipped to three different outlets. The numbers of units of apples and peaches that are shipped to the three outlets are shown in the matrix \(A\). $$ A=\left[\begin{array}{rrr} 125 & 100 & 75 \\ 100 & 175 & 125 \end{array}\right] $$ (a) The profit per unit of apples is \(\$ 3.50\) and the profit per unit of peaches is \(\$ 6\). Organize the profits per unit in a matrix \(B\). (b) Compute \(B A\) and interpret the result.

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{r} x+2 y=0 \\ x+y=6 \\ 3 x-2 y=8 \end{array}\right. $$

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$ \left|\begin{array}{rrrrr} 3 & -2 & 4 & 3 & 1 \\ -1 & 0 & 2 & 1 & 0 \\ 5 & -1 & 0 & 3 & 2 \\ 4 & 7 & -8 & 0 & 0 \\ 1 & 2 & 3 & 0 & 2 \end{array}\right| $$

Use an inverse matrix to solve (if possible) the system of linear equations. $$ \left\\{\begin{aligned} -0.4 x+0.8 y &=1.6 \\ 2 x-4 y &=5 \end{aligned}\right. $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{l} 8 x-4 y=7 \\ 5 x+2 y=1 \end{array}\right. $$

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$ \left|\begin{array}{rrrrr} -2 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & -4 \end{array}\right| $$

Use an inverse matrix to solve (if possible) the system of linear equations. $$ \left\\{\begin{aligned} 0.2 x-0.6 y &=2.4 \\ -x+1.4 y &=-8.8 \end{aligned}\right. $$

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

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

Use an inverse matrix to solve (if possible) the system of linear equations. $$ \left\\{\begin{array}{r} -\frac{1}{4} x+\frac{3}{8} y=-2 \\ \frac{3}{2} x+\frac{3}{4} y=-12 \end{array}\right. $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{c} -x+2 y=1.5 \\ 2 x-4 y=3 \end{array}\right. $$

Evaluate the determinant of the matrix. Do not use a graphing utility. $$ \left[\begin{array}{rrr} 1 & 0 & 0 \\ -4 & -1 & 0 \\ 5 & 1 & 5 \end{array}\right] $$

Use an inverse matrix to solve (if possible) the system of linear equations. $$ \left\\{\begin{array}{l} \frac{5}{6} x-y=-20 \\ \frac{4}{3} x-\frac{7}{2} y=-51 \end{array}\right. $$

Let matrices \(A, B, C\), and \(D\) be of orders \(2 \times 3,2 \times 3,3 \times 2\), and \(2 \times 2\), respectively. Determine whether the matrices are of proper order to perform the operation(s). If so, give the order of the answer. $$ B C $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{l} 2 x-y=-0.1 \\ 3 x+2 y=1.6 \end{array}\right. $$

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

Use an inverse matrix to solve (if possible) the system of linear equations. $$ \left\\{\begin{array}{rr} 4 x-y+z= & -5 \\ 2 x+2 y+3 z= & 10 \\ 5 x-2 y+6 z= & 1 \end{array}\right. $$

Let matrices \(A, B, C\), and \(D\) be of orders \(2 \times 3,2 \times 3,3 \times 2\), and \(2 \times 2\), respectively. Determine whether the matrices are of proper order to perform the operation(s). If so, give the order of the answer. $$ B C-D $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{rr} 2 x+2 y-z= & 2 \\ x-3 y+z= & -28 \\ -x+y= & 14 \end{array}\right. $$

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

Use an inverse matrix to solve (if possible) the system of linear equations. $$ \left\\{\begin{array}{rr} 4 x-2 y+3 z= & -2 \\ 2 x+2 y+5 z= & 16 \\ 8 x-5 y-2 z= & 4 \end{array}\right. $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{rr} -x+y-z= & -14 \\ 2 x-y+z= & 21 \\ 3 x+2 y+z= & 19 \end{array}\right. $$

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

In Exercises 65 and 66, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$ \left\\{\begin{aligned} 7 x-3 y &+2 w=& 41 \\ -2 x+y &-w=&-13 \\ 4 x &+z-2 w=& 12 \\ -x+y &-w=&-8 \end{aligned}\right. $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{l} 2 x+3 z=3 \\ 4 x-3 y+7 z=5 \\ 8 x-9 y+15 z=9 \end{array}\right. $$

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

In Exercises 65 and 66, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$ \left\\{\begin{aligned} 2 x+5 y+w=& 11 \\ x+4 y+2 z-2 w=&-7 \\ 2 x-2 y+5 z+w=& 3 \\ x &-3 w=-1 \end{aligned}\right. $$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} 2 x-y+3 z &=24 \\ 2 y-z &=14 \\ 7 x-5 y &=6 \end{aligned}\right. $$

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

In Exercises 67 and 68, develop for the given matrix a system of equations that has the given solution. Use an inverse matrix to verify that the system of equations has the desired solution. $$ \left[\begin{array}{rrr} 2 & 1 & 3 \\ 4 & 0 & -2 \\ 0 & 3 & 2 \end{array}\right] \quad \begin{aligned} &x=2 \\ &y=-3 \\ &z=5 \end{aligned} $$

Professional athletes frequently have bonus or incentive clauses in their contracts. For example, a defensive football player might receive bonuses for defensive plays such as sacks, interceptions, and/or key tackles. In one contract, a sack is worth $$\$ 2000$$ an interception is worth $$\$ 1000$$ and a key tackle is worth $$\$ 800$$ The table shows the numbers of sacks, interceptions, and key tackles for three players. $$ \begin{array}{|c|c|c|c|} \hline \text { Player } & \text { Sacks } & \text { Interceptions } & \text { Key tackles } \\ \hline \text { Player X } & 3 & 0 & 4 \\ \hline \text { Player Y } & 1 & 2 & 5 \\ \hline \text { Player Z } & 2 & 3 & 3 \\ \hline \end{array} $$ (a) Write a matrix \(D\) that represents the number of each type of defensive play \(i\) made by each player \(j\) using the data from the table. State what each entry \(d_{i j}\) of the matrix represents. (b) Write a matrix \(B\) that represents the bonus amount received for each type of defensive play. State what each entry \(b_{i j}\) of the matrix represents. (c) Find the product \(B D\) of the two matrices and state what each entry of matrix \(B D\) represents. (d) Which player receives the largest bonus?

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{r} x+y-5 z=3 \\ x-2 z=1 \\ 2 x-y-z=1 \end{array}\right. $$

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

In Exercises 65 and 66, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$ \left[\begin{array}{rrr} 1 & 0 & 2 \\ 1 & 1 & 1 \\ 2 & -1 & 0 \end{array}\right] \quad \begin{aligned} &x=5 \\ &y=-2 \\ &z=1 \end{aligned} $$

You are choosing between two monthly long-distance phone plans offered by two different companies. Company A charges $$\$ 0.05$$ per minute for in-state calls, $$\$ 0.12$$ per minute for state-to-state calls, and $$\$ 0.30$$ per minute for international calls. Company \(\mathrm{B}\) charges $$\$ 0.085$$ per minute for in-state calls, $$\$ 0.10$$ per minute for state-to-state calls, and $$\$ 0.25$$ per minute for international calls. In a month, you normally use 20 minutes on in-state calls, 60 minutes on state-to-state calls, and 30 minutes on international calls. (a) Write a matrix \(C\) that represents the charges for each type of call \(i\) by each company \(j\). State what each entry \(c_{i j}\) of the matrix represents. (b) Write a matrix \(T\) that represents the times spent on the phone for each type of call. State what each entry of the matrix represents. (c) Find the product \(T C\) and state what each entry of the matrix represents. (d) Which company should you choose? Explain.

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

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

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. $$ \text { Total investment }=\$ 35,000 \text { ; annual return }=\$ 2950 $$