Chapter 8

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

Determine whether the statement is true or false. Justify your answer. If a square matrix has an entire row of zeros, then the determinant will always be zero.

Find a system of linear equations that has the given solution. (There are many correct answers.) (5,0)

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{array}{l} y-e^{-x}=1 \\ y-\ln x=3 \end{array}\right.$$

Operations with Matrices Use a graphing utility to perform the operations for the matrices \(A, B,\) and \(C\) and the scalar \(c .\) Write a brief statement comparing the results of parts (a) and (b). $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 0 & -2 & 3 \\ 4 & -3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & 3 & 0 \\ 4 & 1 & -2 \\ -1 & 2 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rrr} 3 & -2 & 1 \\ -4 & 0 & 3 \\ -1 & 3 & -2 \end{array}\right], \text { and } c=3 \end{aligned}$$ (a) \(A(B C)\) (b) \((A B) C\)

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{2 x^{3}-x^{2}+x+5}{x^{2}+3 x+2}$$

Use matrices to solve the system of equations, if possible. Use Gauss-Jordan elimination. $$\left\\{\begin{array}{c} x+y-5 z=3 \\ x-2 z=1 \\ 2 x-y-z=0 \end{array}\right.$$

Explain how to determine whether the inverse of a \(2 \times 2\) matrix exists. If so, explain how to find the inverse.

Determine whether the statement is true or false. Justify your answer. If two columns of a square matrix are the same, then the determinant of the matrix will be zero.

Find a system of linear equations that has the given solution. (There are many correct answers.) (-6,1)

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{aligned} 2 \ln x+y &=4 \\ e^{x}-y &=0 \end{aligned}\right.$$

Operations with Matrices Use a graphing utility to perform the operations for the matrices \(A, B,\) and \(C\) and the scalar \(c .\) Write a brief statement comparing the results of parts (a) and (b). $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 0 & -2 & 3 \\ 4 & -3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & 3 & 0 \\ 4 & 1 & -2 \\ -1 & 2 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rrr} 3 & -2 & 1 \\ -4 & 0 & 3 \\ -1 & 3 & -2 \end{array}\right], \text { and } c=3 \end{aligned}$$ (a) \(c(A B)\) (b) \((c A) B\)

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{x^{3}+2 x^{2}-x+1}{x^{2}+3 x-4}$$

Use matrices to solve the system of equations, if possible. Use Gauss-Jordan elimination. $$\left\\{\begin{aligned} 2 x+2 y-z &=2 \\ x-3 y+z &=28 \\ -x+y &=14 \end{aligned}\right.$$

Determine whether the statement is true or false. Justify your answer. Exploration Find a pair of \(3 \times 3\) matrices \(A\) and \(B\) to demonstrate that \(|A+B| \neq|A|+|B|\)

Find a system of linear equations that has the given solution. (There are many correct answers.) (2.5,-4)

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{array}{l} y=x^{3}-2 x^{2}+1 \\ y=1-x^{2} \end{array}\right.$$

Operations with Matrices Use a graphing utility to perform the operations for the matrices \(A, B,\) and \(C\) and the scalar \(c .\) Write a brief statement comparing the results of parts (a) and (b). $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 0 & -2 & 3 \\ 4 & -3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & 3 & 0 \\ 4 & 1 & -2 \\ -1 & 2 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rrr} 3 & -2 & 1 \\ -4 & 0 & 3 \\ -1 & 3 & -2 \end{array}\right], \text { and } c=3 \end{aligned}$$ (a) \((A+B)^{2}\) (b) \(A^{2}+A B+B A+B^{2}\)

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{x^{4}}{(x-1)^{3}}$$

Explain in your own words how to write a system of three linear equations in three variables as a matrix equation \(A X=B,\) as well as how to solve the system using an inverse matrix.

Determine whether the statement is true or false. Justify your answer. Think About It Let \(A\) be a \(3 \times 3\) matrix such that \(|A|=5 .\) Can you use this information to find \(|2 A| ?\) Explain.

Find a system of linear equations that has the given solution. (There are many correct answers.) \(\left(-\frac{3}{4}, 12\right)\)

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{array}{l} y=x^{3}-2 x^{2}+x-1 \\ y=-x^{2}+3 x-1 \end{array}\right.$$

Operations with Matrices Use a graphing utility to perform the operations for the matrices \(A, B,\) and \(C\) and the scalar \(c .\) Write a brief statement comparing the results of parts (a) and (b). $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 0 & -2 & 3 \\ 4 & -3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & 3 & 0 \\ 4 & 1 & -2 \\ -1 & 2 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rrr} 3 & -2 & 1 \\ -4 & 0 & 3 \\ -1 & 3 & -2 \end{array}\right], \text { and } c=3 \end{aligned}$$ (a) \((A-B)^{2}\) (b) \(A^{2}-A B-B A+B^{2}\)

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{4 x^{4}}{(2 x-1)^{3}}$$

Use matrices to solve the system of equations, if possible. Use Gauss-Jordan elimination. $$\left\\{\begin{array}{c} x+y+4 z=5 \\ 2 x+y-z=9 \end{array}\right.$$

If \(A\) is a \(2 \times 2\) matrix given by \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right],\) then \(A\) is invertible if and only if \(a d-b c \neq 0 .\) If \(a d-b c \neq 0,\) verify that the inverse is \(A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right]\).

(A) find the determinant of \(A,\) (b) find \(A^{-1},\) (c) find \(\operatorname{det}\left(A^{-1}\right),\) and (d) compare your results from parts (a) and (c). Make a conjecture based on your results. $$A=\left[\begin{array}{rr} 1 & 2 \\ -2 & 2 \end{array}\right]$$

Find the point of equilibrium of the demand and supply equations. The point of equilibrium is the price \(p\) and the number of units \(x\) that satisfy both the demand and supply equations. Demand \(\quad\) Supply \(p=500-0.4 x \quad p=380+0.1 x\)

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{aligned} x y-1 &=0 \\ -5 x-2 y+1 &=0 \end{aligned}\right.$$

Operations with Matrices Perform the operations (a) using a graphing utility and (b) by hand algebraically. If it is not possible to perform the operation(s), state the reason. $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -1 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 4 & -1 \\ -2 & -1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rr} 1 & 2 \\ -2 & 3 \\ 1 & 0 \end{array}\right], \quad c=-2, \text { and } d=-3 \end{aligned}$$ $$A+c B$$

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{x}{x^{3}-x^{2}-2 x+2}$$

Use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. $$\left\\{\begin{array}{rr} x+y+4 z= & 2 \\ 2 x+5 y+20 z= & 10 \\ -x+3 y+8 z= & -2 \end{array}\right.$$

Consider matrices of the form \(A=\left[\begin{array}{cccccc}a_{11} & 0 & 0 & 0 & . . & 0 \\ 0 & a_{22} & 0 & 0 & . . & 0 \\ 0 & 0 & a_{33} & 0 & . . & 0 \\ \vdots & \vdots & \vdots & \vdots & . . & \vdots \\ 0 & 0 & 0 & 0 & . . & a_{n n}\end{array}\right]\) (a) Write a \(2 \times 2\) matrix and a \(3 \times 3\) matrix in the form of \(A .\) Find the inverse of each. (b) Use the result of part (a) to make a conjecture about the inverse of a matrix in the form of \(A\).

(A) find the determinant of \(A,\) (b) find \(A^{-1},\) (c) find \(\operatorname{det}\left(A^{-1}\right),\) and (d) compare your results from parts (a) and (c). Make a conjecture based on your results. $$A=\left[\begin{array}{ll} 5 & -1 \\ 2 & -1 \end{array}\right]$$

Find the point of equilibrium of the demand and supply equations. The point of equilibrium is the price \(p\) and the number of units \(x\) that satisfy both the demand and supply equations. Demand \(\quad\) Supply \(p=100-0.05 x \quad p=25+0.1 x\)

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{aligned} x y-2 &=0 \\ 3 x-2 y+4 &=0 \end{aligned}\right.$$

Operations with Matrices Perform the operations (a) using a graphing utility and (b) by hand algebraically. If it is not possible to perform the operation(s), state the reason. $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -1 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 4 & -1 \\ -2 & -1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rr} 1 & 2 \\ -2 & 3 \\ 1 & 0 \end{array}\right], \quad c=-2, \text { and } d=-3 \end{aligned}$$ $$B+d A$$

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{2 x^{2}+x+8}{x^{4}+8 x^{2}+16}$$

Use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. $$\left\\{\begin{aligned} x+y+z &=0 \\ 2 x+3 y+z &=0 \\ 3 x+5 y+z &=0 \end{aligned}\right.$$

Let \(A\) be a \(2 \times 2\) matrix given by $$ A=\left[\begin{array}{ll} x & 0 \\ 0 & y \end{array}\right] $$ Use the determinant of \(A\) to determine the conditions under which \(A^{-1}\) exists.

(A) find the determinant of \(A,\) (b) find \(A^{-1},\) (c) find \(\operatorname{det}\left(A^{-1}\right),\) and (d) compare your results from parts (a) and (c). Make a conjecture based on your results. $$A=\left[\begin{array}{rrr} 1 & -3 & -2 \\ -1 & 3 & 1 \\ 0 & 2 & -2 \end{array}\right]$$

Find the point of equilibrium of the demand and supply equations. The point of equilibrium is the price \(p\) and the number of units \(x\) that satisfy both the demand and supply equations. Demand \(\quad\) Supply \(p=140-0.00002 x \quad p=80+0.00001 x\)

Use a graphing utility to graph the cost and revenue functions in the same viewing window. Find the sales \(x\) necessary to break even \((R=C)\) and the corresponding revenue \(R\) obtained by selling \(x\) units. (Round to the nearest whole unit.) Cost \(C=8650 x+250,000\) Revenue \(R=9950 x\)

Operations with Matrices Perform the operations (a) using a graphing utility and (b) by hand algebraically. If it is not possible to perform the operation(s), state the reason. $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -1 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 4 & -1 \\ -2 & -1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rr} 1 & 2 \\ -2 & 3 \\ 1 & 0 \end{array}\right], \quad c=-2, \text { and } d=-3 \end{aligned}$$ $$c(A B)$$

Write the partial fraction decomposition for the rational function. Identify the graph of the rational function and the graph of each term of its decomposition. State any relationship between the vertical asymptotes of the rational function and the vertical asymptotes of the terms of the decomposition. $$y=\frac{x-12}{x(x-4)}$$

Use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. $$\left\\{\begin{array}{rr} 2 x+10 y+2 z= & 6 \\ x+5 y+2 z= & 6 \\ x+5 y+z= & 3 \\ -3 x-15 y-3 z= & -9 \end{array}\right.$$

Solve the equation algebraically. Round your result to three decimal places. $$e^{2 x}+2 e^{x}-15=0$$

(A) find the determinant of \(A,\) (b) find \(A^{-1},\) (c) find \(\operatorname{det}\left(A^{-1}\right),\) and (d) compare your results from parts (a) and (c). Make a conjecture based on your results. $$A=\left[\begin{array}{rrr} -1 & 3 & 2 \\ 1 & 3 & -1 \\ 1 & 1 & -2 \end{array}\right]$$

Find the point of equilibrium of the demand and supply equations. The point of equilibrium is the price \(p\) and the number of units \(x\) that satisfy both the demand and supply equations. Demand \(\quad\) Supply \(p=400-0.0002 x \quad p=225+0.0005 x\)