Chapter 9

Use Cramer's Rule to solve each system. $$\left\\{\begin{aligned}x-3 y+z &=-2 \\\x+2 y &=8 \\\2 x-y &=1\end{aligned}\right.$$

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{ll} {4} & {2} \\ {6} & {1} \\ {3} & {5} \end{array}\right], \quad B=\left[\begin{array}{rrr} {2} & {3} & {4} \\ {-1} & {-2} & {0} \end{array}\right] $$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{l} {2 x+2 y+7 z=-1} \\ {2 x+y+2 z=2} \\ {4 x+6 y+z=15} \end{array}\right. $$

Use Cramer's Rule to solve each system. $$\left\\{\begin{array}{l}{x+y+z=4} \\\\{x-2 y+z=7} \\\\{x+3 y+2 z=4}\end{array}\right.$$

Write each matrix equation as a system of linear equations without matrices. $$ \left[\begin{array}{cc} {4} & {-7} \\ {2} & {-3} \end{array}\right]\left[\begin{array}{l} {x} \\ {y} \end{array}\right]=\left[\begin{array}{r} {-3} \\ {1} \end{array}\right] $$

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{ll} {2} & {4} \\ {3} & {1} \\ {4} & {2} \end{array}\right], \quad B=\left[\begin{array}{rrr} {3} & {2} & {0} \\ {-1} & {-3} & {5} \end{array}\right] $$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{l} {3 x+2 y+3 z=3} \\ {4 x-5 y+7 z=1} \\ {2 x+3 y-2 z=6} \end{array}\right. $$

Write each matrix equation as a system of linear equations without matrices. $$ \left[\begin{array}{rr} {3} & {0} \\ {-3} & {1} \end{array}\right]\left[\begin{array}{l} {x} \\ {y} \end{array}\right]=\left[\begin{array}{r} {6} \\ {-7} \end{array}\right] $$

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{rrrr} {2} & {-3} & {1} & {-1} \\ {1} & {1} & {-2} & {1} \end{array}\right], \quad B=\left[\begin{array}{rr} {1} & {2} \\ {-1} & {1} \\ {5} & {4} \\ {10} & {5} \end{array}\right] $$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} w+x+y+z &=4 \\ 2 w+x-2 y-z &=0 \\ w-2 x-y-2 z &=-2 \\ 3 w+2 x+y+3 z &=4 \end{aligned}\right. $$

Use Cramer's Rule to solve each system. $$\left\\{\begin{aligned}x+& 2 z=4 \\\2 y-z &=5 \\\2 x+3 y &=13\end{aligned}\right.$$

Write each matrix equation as a system of linear equations without matrices. $$ \left[\begin{array}{rrr} {2} & {0} & {-1} \\ {0} & {3} & {0} \\ {1} & {1} & {0} \end{array}\right]\left[\begin{array}{l} {x} \\ {y} \\ {z} \end{array}\right]=\left[\begin{array}{l} {6} \\ {9} \\ {5} \end{array}\right] $$

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{rrrr} {2} & {-1} & {3} & {2} \\ {1} & {0} & {-2} & {1} \end{array}\right], \quad B=\left[\begin{array}{rr} {-1} & {2} \\ {1} & {1} \\ {3} & {-4} \\ {6} & {5} \end{array}\right] $$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{c} {w+x+y+z=5} \\ {w+2 x-y-2 z=-1} \\ {w-3 x-3 y-z=-1} \\ {2 w-x+2 y-z=-2} \end{array}\right. $$

Use Cramer's Rule to solve each system. $$\left\\{\begin{array}{l}{3 x+2 z=4} \\\\{5 x-y=-4} \\\\{4 y+3 z=22}\end{array}\right.$$

Write each matrix equation as a system of linear equations without matrices. $$ \left[\begin{array}{rrr} {-1} & {0} & {1} \\ {0} & {-1} & {0} \\ {0} & {1} & {1} \end{array}\right]\left[\begin{array}{l} {x} \\ {y} \\ {z} \end{array}\right]=\left[\begin{array}{r} {-4} \\ {2} \\ {4} \end{array}\right] $$

Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ 4 B-3 C $$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} 3 w-4 x+y+z &=9 \\ w+x-y-z &=0 \\ 2 w+x+4 y-2 z &=3 \\ -w+2 x+y-3 z &=3 \end{aligned}\right. $$

Evaluate each determinant. $$\left|\begin{array}{rrrr}{4} & {2} & {8} & {-7} \\\\{-2} & {0} & {4} & {1} \\\\{5} & {0} & {0} & {5} \\\\{4} & {0} & {0} & {-1}\end{array}\right|$$

A. Write each linear system as a matrix equation in the form \(A X=B\) B. Solve the system using the inverse that is given for the coefficient matrix. $$ \left\\{\begin{array}{l} {2 x+6 y+6 z=8} \\ {2 x+7 y+6 z=10} \\ {2 x+7 y+7 z=9} \end{array}\right. $$

Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ 5 C-2 B $$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} 2 w+y-3 z &=8 \\ w-x+4 z &=-10 \\ 3 w+5 x-y-z &=20 \\ w+x-y-z &=6 \end{aligned}\right. $$

Describe what happens when Gaussian elimination is used to solve a system with dependent equations.

Evaluate each determinant. $$\left|\begin{array}{rrrr}{3} & {-1} & {1} & {2} \\\\{-2} & {0} & {0} & {0} \\\\{2} & {-1} & {-2} & {3} \\\\{1} & {4} & {2} & {3}\end{array}\right|$$

A. Write each linear system as a matrix equation in the form \(A X=B\) B. Solve the system using the inverse that is given for the coefficient matrix. $$ \left\\{\begin{array}{r} {x+2 y+5 z=2} \\ {2 x+3 y+8 z=3} \\ {-x+y+2 z=3} \end{array}\right. $$

Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ B C+C B $$

Find the quadratic function \(f(x)=a x^{2}+b x+c\) for which \(f(-2)=-4, f(1)=2,\) and \(f(2)=0\)

In solving a system of dependent equations in three variables, one student simply said that there are infinitely many solutions. A second student expressed the solution set as \(\\{(4 z+3,5 z-1, z)\\} .\) Which is the better form of expressing the solution set and why?

Evaluate each determinant. $$\left|\begin{array}{rrrr}{-2} & {-3} & {3} & {5} \\\\{1} & {-4} & {0} & {0} \\\\{1} & {2} & {2} & {-3} \\\\{2} & {0} & {1} & {1}\end{array}\right|$$

A. Write each linear system as a matrix equation in the form \(A X=B\) B. Solve the system using the inverse that is given for the coefficient matrix. $$ \left\\{\begin{aligned} x-y+z &=8 \\ 2 y-z &=-7 \\ 2 x+3 y &=1 \end{aligned}\right. $$

Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ A(B+C) $$

Find the quadratic function \(f(x)=a x^{2}+b x+c\) for which \(f(-1)=5, f(1)=3,\) and \(f(2)=5\)

a. The figure shows the intersections of a number of one-way streets. The numbers given represent traffic flow at a peak period (from 4 \(\mathrm{p}\). M. to 5: 30 \(\mathrm{PM}\).). Use the figure to write a linear system of six equations in seven variables based on the idea that at each intersection the number of cars entering must equal the number of cars leaving. b. Use a graphing utility with a [ ref ]or [\mathrm{ rref } ] \text { command to } find the complete solution to the system. (b) Consider the system of linear equation, \\[ \begin{aligned} x_{4}-x_{1} &=200 \\ x_{5}+x_{2}-x_{4} &=100 \\ x_{5}+x_{3} &=700 \\ x_{3}+x_{7} &=900 \\ x_{1}-x_{6} &=100 \\ x_{6}-x_{2}-x_{7} &=-600 \end{aligned} \\]

Evaluate each determinant. $$\left|\begin{array}{rrrr}{1} & {-3} & {2} & {0} \\\\{-3} & {-1} & {0} & {-2} \\\\{2} & {1} & {3} & {1} \\\\{2} & {0} & {-2} & {0}\end{array}\right|$$

a. Write each linear system as a matrix equation in the form \(A X=B\) b. Solve the system using the inverse that is given for the coefficient matrix. $$\left\\{\begin{aligned} x-6 y+3 z &=11 \\ 2 x-7 y+3 z &=14 \\ 4 x-12 y+5 z &=25 \end{aligned}\right.$$

Find the cubic function \(f(x)=a x^{3}+b x^{2}+c x+d\) for which \(f(-1)=0, f(1)=2, f(2)=3,\) and \(f(3)=12\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. $$ \begin{aligned} &\text { I omitted row 3 from }\left[\begin{array}{rrr|r} {1} & {-1} & {-2} & {2} \\ {0} & {1} & {-10} & {-1} \\ {0} & {0} & {0} & {5} \end{array}\right] \text { and expressed the }\\\ &\text { system in the form }\left[\begin{array}{rrr|r} {1} & {-1} & {-2} & {2} \\ {0} & {1} & {-10} & {-1} \end{array}\right] \end{aligned} $$

Evaluate each determinant. $$\left|\begin{array}{rr}{\left|\begin{array}{rr}{3} & {1} \\\\{-2} & {3}\end{array}\right|} & {\left|\begin{array}{rr}{7} & {0} \\\\{1} & {5}\end{array}\right|} \\\\{\left|\begin{array}{rr}{3} & {0} \\\\{0} & {7}\end{array}\right|} & {\left|\begin{array}{rr}{9} & {-6} \\\\{3} & {5}\end{array}\right|}\end{array}\right|$$

Find the cubic function \(f(x)=a x^{3}+b x^{2}+c x+d\) for which \(f(-1)=3, f(1)=1, f(2)=6,\) and \(f(3)=7\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. $$ \begin{aligned} &\text { I omitted row 3 from }\left[\begin{array}{rrr|r} {1} & {-1} & {-2} & {2} \\ {0} & {1} & {-10} & {-1} \\ {0} & {0} & {0} & {0} \end{array}\right] \text { and expressed }\\\ &\text { the system in the form }\left[\begin{array}{rrr|r} {1} & {-1} & {-2} & {2} \\ {0} & {1} & {-10} & {-1} \end{array}\right] \end{aligned} $$

Evaluate each determinant. $$Therefore, the given determinant evaluates to -407 . \left|\begin{array}{rr} {\left|\begin{array}{rr} {5} & {0} \\ {4} & {-3} \end{array}\right|} & {\left|\begin{array}{rr} {-1} & {0} \\ {0} & {-1} \end{array}\right|} \\ {\left|\begin{array}{rr} {7} & {-5} \\ {4} & {6} \end{array}\right|} & {\left|\begin{array}{rr} {4} & {1} \\ {-3} & {5} \end{array}\right|} \end{array}\right| $$

Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ A(B C) $$

Solve the system: $$ \left\\{\begin{array}{c} {2 \ln w+\ln x+3 \ln y-2 \ln z=-6} \\ {4 \ln w+3 \ln x+\ln y-\ln z=-2} \\ {\ln w+\ln x+\ln y+\ln z=-5} \\ {\ln w+\ln x-\ln y-\ln z=5} \end{array}\right. $$ (Hint: Let \(A=\ln w, B=\ln x, C=\ln y,\) and \(D=\ln z .\) Solve the system for \(A, B, C,\) and \(D .\) Then use the logarithmic equations to find \(w, x, y, \text { and } z .)\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I solved a nonsquare system in which the number of equations was the same as the number of variables.

Write the system of linear equations for which Cramer's Rule yields the given determinants. $$ D=\left|\begin{array}{rr} {2} & {-4} \\ {3} & {5} \end{array}\right|, \quad D_{x}=\left|\begin{array}{rr} {8} & {-4} \\ {-10} & {5} \end{array}\right| $$

Find \(A^{-1}\) and check. $$ A=\left[\begin{array}{cc} {e^{x}} & {e^{3 x}} \\ {-e^{3 x}} & {e^{5 x}} \end{array}\right] $$

Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ A(C B) $$

Solve the system: $$ \left\\{\begin{aligned} \ln w+\ln x+\ln y+& \ln z &=-1 \\ -\ln w+4 \ln x+\ln y-& \ln z=& 0 \\ \ln w-2 \ln x+\ln y-& 2 \ln z=& 11 \\ -\ln w-2 \ln x+\ln y+& 2 \ln z=&-3 \end{aligned}\right. $$ (Hint: Let \(A=\ln w, B=\ln x, C=\ln y,\) and \(D=\ln z .\) Solve the system for \(A, B, C,\) and \(D .\) Then use the logarithmic equations to find \(w, x, y, \text { and } z .)\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Models for controlling traffic flow are based on an equal number of cars entering an intersection and leaving that intersection,

Write the system of linear equations for which Cramer's Rule yields the given determinants. $$D=\left|\begin{array}{rr}{2} & {-3} \\\\{5} & {6}\end{array}\right|, \quad D_{x}=\left|\begin{array}{rr} {8} & {-3} \\\\{11} & {6}\end{array}\right|$$