Chapter 6

Represent the system of linear equations in the form \(A X=B\) \(4 x-3 y+2 z=8\) \(-x+4 y+3 z=2\) \(-2 x \quad-5 z=2\)

Solve the system, if possible. $$ \begin{aligned} 2 x-y-z &=0 \\ x-y-z &=-2 \\ 3 x-2 y-2 z &=-2 \end{aligned} $$

If possible, solve the system of linear equations and check your answer. $$ \begin{array}{r} -2 x-y=-2 \\ 3 x+4 y=-7 \end{array} $$

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rr}-3 & 1 \\\2 & -4\end{array}\right], \quad B=\left[\begin{array}{rrr}1 & 0 & -2 \\\\-4 & 8 & 1\end{array}\right]$$

Choose two matrices \(A\) and \(B\) with dimension \(2 \times 2\). Calculate det \(A\), det \(B\), and \(\operatorname{det}(A B)\). Repeat this process until you are able to discover how these three determinants are related. Summarize your results.

Represent the system of linear equations in the form \(A X=B\) \(\begin{aligned} 4 x-y+3 z &=-2 \\ x+2 y+5 z &=11 \\ 2 x-3 y &=-1 \end{aligned}\)

Solve the system, if possible. $$ \begin{array}{rr} x-4 y+3 z= & 26 \\ -x+3 y-2 z= & -19 \\ -y+z= & 10 \end{array} $$

If possible, solve the system of linear equations and check your answer. $$ \begin{aligned} &2 x-9 y=-17\\\ &8 x+5 y=14 \end{aligned} $$

Calculate det \(A\) and det \(A^{-1}\) for different matrices. Compare the determinants. Try to generalize your results.

Represent the system of linear equations in the form \(A X=B\) \(\begin{aligned} x-2 y+z &=12 \\ 4 y+3 z &=13 \\\\-2 x+7 y &=-2 \end{aligned}\)

Solve the system, if possible. $$ \begin{aligned} &4 x-y-z=0\\\ &4 x-2 y \quad=0\\\ &2 x \quad+z=1 \end{aligned} $$

If possible, solve the system of linear equations and check your answer. $$ \begin{aligned} &3 x+6 y=0\\\ &4 x-2 y=-5 \end{aligned} $$

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rrr}1 & 0 & -2 \\\3 & -4 & 1 \\\2 & 0 & 5\end{array}\right], \quad B=\left[\begin{array}{r}1 \\\\-1 \\\3\end{array}\right]$$

Complete the following. (A) Write the system in the form \(A X=B\). (B) Solve the system by finding \(A^{-1}\) and then using the equation \(\boldsymbol{X}=\boldsymbol{A}^{-1} \boldsymbol{B}\). (Hint: Some of your answers from Exercises \(15-28\) may be helpful.) $$ \begin{aligned} &x+2 y=3\\\ &x+3 y=6 \end{aligned} $$

Solve the system, if possible. $$ \begin{array}{r} 5 x & +4 z=7 \\ 2 x-4 y=6 \\ 3 y+3 z=3 \end{array} $$

If possible, solve the system of linear equations and check your answer. $$ \begin{aligned} \frac{1}{2} x-y &=-5 \\ x+\frac{1}{2} y &=10 \end{aligned} $$

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rrrr}1 & -1 & 3 & -2 \\\1 & 0 & 3 & 4 \\\2 & -2 & 0 & 8 \end{array}\right], \quad B=\left[\begin{array}{rr}1 & -1 \\\0 & 5 \\\2 & 3 \\\\-5 & 4\end{array}\right]$$

Complete the following. (A) Write the system in the form \(A X=B\). (B) Solve the system by finding \(A^{-1}\) and then using the equation \(\boldsymbol{X}=\boldsymbol{A}^{-1} \boldsymbol{B}\). (Hint: Some of your answers from Exercises \(15-28\) may be helpful.) $$ \begin{array}{l} 2 x+y=4 \\ -x+2 y=-1 \end{array} $$

Solve the system, if possible. $$y+2 z=-5$$ $$3 x \quad-2 z=-6$$ $$-x-4 y \quad=11$$

If possible, solve the system of linear equations and check your answer. $$ \begin{array}{lr} -x-\frac{1}{3} y= & -4 \\ \frac{1}{3} x+2 y= & 7 \end{array} $$

Use the given \(A\) and \(B\) to evaluate each expression. $$A=\left[\begin{array}{rrr}3 & -2 & 4 \\\5 & 2 & -3 \\\7 & 5 & 4\end{array}\right], B=\left[\begin{array}{rrr}1 & 1 & -5 \\\\-1 & 0 & -7 \\\\-6 & 4 & 3\end{array}\right]$$ $$A B$$

Complete the following. (A) Write the system in the form \(A X=B\). (B) Solve the system by finding \(A^{-1}\) and then using the equation \(\boldsymbol{X}=\boldsymbol{A}^{-1} \boldsymbol{B}\). (Hint: Some of your answers from Exercises \(15-28\) may be helpful.) $$ \begin{array}{rr} -x+2 y= & 5 \\ 3 x-5 y= & -2 \end{array} $$

Solve the system, if possible. $$ \begin{array}{rr} 5 x-2 y+z= & 5 \\ x+y-2 z= & -2 \\ 4 x-3 y+3 z= & 7 \end{array} $$

If possible, solve the system of linear equations and check your answer. $$ \begin{array}{rr} 3 x-2 y= & 5 \\ -6 x+4 y= & -10 \end{array} $$

Use the given \(A\) and \(B\) to evaluate each expression. $$A=\left[\begin{array}{rrr}3 & -2 & 4 \\\5 & 2 & -3 \\\7 & 5 & 4\end{array}\right], B=\left[\begin{array}{rrr}1 & 1 & -5 \\\\-1 & 0 & -7 \\\\-6 & 4 & 3\end{array}\right]$$ $$B A$$

Complete the following. (A) Write the system in the form \(A X=B\). (B) Solve the system by finding \(A^{-1}\) and then using the equation \(\boldsymbol{X}=\boldsymbol{A}^{-1} \boldsymbol{B}\). (Hint: Some of your answers from Exercises \(15-28\) may be helpful.) $$ \begin{array}{c} x+3 y=-3 \\ 2 x+5 y=-2 \end{array} $$

Solve the system, if possible. $$ \begin{aligned} 2 x-4 y-z &=2 \\ x+y-3 z &=10 \\ -x-7 y+8 z &=2 \end{aligned} $$

If possible, solve the system of linear equations and check your answer. $$ \begin{aligned} &\frac{1}{2} x-\frac{3}{4} y=\frac{1}{2}\\\ &\frac{1}{5} x-\frac{3}{10} y=\frac{1}{5} \end{aligned} $$

Use the given \(A\) and \(B\) to evaluate each expression. $$A=\left[\begin{array}{rrr}3 & -2 & 4 \\\5 & 2 & -3 \\\7 & 5 & 4\end{array}\right], B=\left[\begin{array}{rrr}1 & 1 & -5 \\\\-1 & 0 & -7 \\\\-6 & 4 & 3\end{array}\right]$$ $$3 A^{2}+2 B$$

Complete the following. (A) Write the system in the form \(A X=B\). (B) Solve the system by finding \(A^{-1}\) and then using the equation \(\boldsymbol{X}=\boldsymbol{A}^{-1} \boldsymbol{B}\). (Hint: Some of your answers from Exercises \(15-28\) may be helpful.) $$ \begin{array}{rr} x+\quad z= & -7 \\ 2 x+y+3 z= & -13 \\ -x+y+z= & -4 \end{array} $$

(Refer to Example \(6 .\) ) The augmented matrix is in reduced row-echelon form and represents a system of linear equations. If possible, solve the system. $$ \left[\begin{array}{ll|r} 1 & 0 & 12 \\ 0 & 1 & 3 \end{array}\right] $$

If possible, solve the system of linear equations and check your answer. $$ \begin{array}{r} 2 x-7 y=8 \\ -3 x+\frac{21}{2} y=5 \end{array} $$

Use the given \(A\) and \(B\) to evaluate each expression. $$A=\left[\begin{array}{rrr}3 & -2 & 4 \\\5 & 2 & -3 \\\7 & 5 & 4\end{array}\right], B=\left[\begin{array}{rrr}1 & 1 & -5 \\\\-1 & 0 & -7 \\\\-6 & 4 & 3\end{array}\right]$$ $$B^{2}-3 A$$

Complete the following. (A) Write the system in the form \(A X=B\). (B) Solve the system by finding \(A^{-1}\) and then using the equation \(\boldsymbol{X}=\boldsymbol{A}^{-1} \boldsymbol{B}\). (Hint: Some of your answers from Exercises \(15-28\) may be helpful.) $$ \begin{aligned} -2 x+y &=-5 \\ x &+z=-5 \\ -x+y &=-4 \end{aligned} $$

If possible, solve the system of linear equations and check your answer. $$ \begin{aligned} 0.6 x-0.2 y &=2 \\ -1.2 x+0.4 y &=3 \end{aligned} $$

Properties of Matrices Use a graphing calculator to evaluate the expression with the given matrices \(A, B,\) and \(C .\) Compare your answers for parts (a) and (b). Then interpret the results. $$A=\left[\begin{array}{rrr}2 & -1 & 3 \\\1 & 3 & -5 \\\0 & -2 & 1\end{array}\right], B=\left[\begin{array}{rrr}6 & 2 & 7 \\\3 & -4 & -5 \\\7 & 1 & 0\end{array}\right]$$ $$C=\left[\begin{array}{lll}1 & 4 & -3 \\\8 & 1 & -1 \\\4 & 6 & -2\end{array}\right]$$ (a) \(A(B+C)\) (b) \(A B+A C\)

Find the minimum value of \(C=4 x+2 y\) subject to the following constraints. $$ \begin{aligned} &\begin{array}{r} x+y \geq 3 \\ 2 x+3 y \leq 12 \end{array}\\\ &x \geq 0, y \geq 0 \end{aligned} $$

Complete the following. (A) Write the system in the form \(A X=B\). (B) Solve the system by finding \(A^{-1}\) and then using the equation \(\boldsymbol{X}=\boldsymbol{A}^{-1} \boldsymbol{B}\). (Hint: Some of your answers from Exercises \(15-28\) may be helpful.) $$ \begin{array}{rr} x+2 y-z= & 2 \\ 2 x+5 y & =-1 \\ -x-y+2 z= & 0 \end{array} $$

If possible, solve the system of linear equations and check your answer. $$ \begin{aligned} &0.2 x-0.1 y=0.5\\\ &0.4 x+0.3 y=2.5 \end{aligned} $$

Properties of Matrices Use a graphing calculator to evaluate the expression with the given matrices \(A, B,\) and \(C .\) Compare your answers for parts (a) and (b). Then interpret the results. $$A=\left[\begin{array}{rrr}2 & -1 & 3 \\\1 & 3 & -5 \\\0 & -2 & 1\end{array}\right], B=\left[\begin{array}{rrr}6 & 2 & 7 \\\3 & -4 & -5 \\\7 & 1 & 0\end{array}\right]$$ $$C=\left[\begin{array}{lll}1 & 4 & -3 \\\8 & 1 & -1 \\\4 & 6 & -2\end{array}\right]$$ (a) \((A-B) C\) (b) \(A C-B C\)

Find the maximum value of \(P=3 x+5 y\) subject to the following constraints. $$ \begin{array}{r} 3 x+y \leq 8 \\ x+3 y \leq 8 \\ x \geq 0, y \geq 0 \end{array} $$

If possible, solve the system of linear equations and check your answer. $$ \begin{aligned} &100 x+200 y=300\\\ &200 x+100 y=0 \end{aligned} $$

Properties of Matrices Use a graphing calculator to evaluate the expression with the given matrices \(A, B,\) and \(C .\) Compare your answers for parts (a) and (b). Then interpret the results. $$A=\left[\begin{array}{rrr}2 & -1 & 3 \\\1 & 3 & -5 \\\0 & -2 & 1\end{array}\right], B=\left[\begin{array}{rrr}6 & 2 & 7 \\\3 & -4 & -5 \\\7 & 1 & 0\end{array}\right]$$ $$C=\left[\begin{array}{lll}1 & 4 & -3 \\\8 & 1 & -1 \\\4 & 6 & -2\end{array}\right]$$ (a) \((A-B)^{2}\) (b) \(A^{2}-A B-B A+B^{2}\)

If possible, maximize and minimize \(z\) subject to the given constraints. $$ z=7 x+6 y $$ $$ \begin{array}{r} x+y \leq 8 \\ x+y \geq 4 \\ x \geq 0, y \geq 0 \end{array} $$

Complete the following. (A) Write the system in the form \(A X=B\). (B) Solve the system by finding \(A^{-1}\) and then using the equation \(\boldsymbol{X}=\boldsymbol{A}^{-1} \boldsymbol{B}\). (Hint: Some of your answers from Exercises \(15-28\) may be helpful.) $$ \begin{array}{r} 2 x-2 y+z=1 \\ x+3 y+2 z=3 \\ 4 x-2 y+4 z=4 \end{array} $$

If possible, solve the nonlinear system of equations. $$ \begin{aligned} x^{2}-y &=0 \\ 2 x+y &=0 \end{aligned} $$

Properties of Matrices Use a graphing calculator to evaluate the expression with the given matrices \(A, B,\) and \(C .\) Compare your answers for parts (a) and (b). Then interpret the results. $$A=\left[\begin{array}{rrr}2 & -1 & 3 \\\1 & 3 & -5 \\\0 & -2 & 1\end{array}\right], B=\left[\begin{array}{rrr}6 & 2 & 7 \\\3 & -4 & -5 \\\7 & 1 & 0\end{array}\right]$$ $$C=\left[\begin{array}{lll}1 & 4 & -3 \\\8 & 1 & -1 \\\4 & 6 & -2\end{array}\right]$$ (a) \((A B) C\) (b) \(A(B C)\)

If possible, maximize and minimize \(z\) subject to the given constraints. $$ z=8 x+3 y $$ $$ \begin{array}{l} 4 x+y \geq 12 \\ x+2 y \geq 6 \\ x \geq 0, y \geq 0 \end{array} $$

Complete the following for the given system of linear equations. (a) Write the system in the form \(A X=B\). (b) Solve the linear system by computing \(X=A^{-1} B\) with a calculator. Approximate the solution to the $$ \begin{array}{rr} 31 x+18 y= & 64.1 \\ 5 x-23 y= & -59.6 \end{array} $$

(Refer to Example \(6 .\) ) The augmented matrix is in reduced row-echelon form and represents a system of linear equations. If possible, solve the system. $$ \left[\begin{array}{lll|r} 1 & 0 & 1 & -2 \\ 0 & 1 & 3 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right] $$