Chapter 9

Find \(A^{-1}\) and check. $$ A=\left[\begin{array}{cc} {e^{2 x}} & {-e^{x}} \\ {e^{3 x}} & {e^{2 x}} \end{array}\right] $$

Let $$ \begin{aligned} A &=\left[\begin{array}{cc} {1} & {0} \\ {0} & {1} \end{array}\right], \quad B=\left[\begin{array}{rr} {1} & {0} \\ {0} & {-1} \end{array}\right], \quad C=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {1} \end{array}\right] \\ D &=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {-1} \end{array}\right] \end{aligned} $$ Find the product of the sum of A and B and the difference between C and D.

Consider the linear system. $$ \left\\{\begin{aligned} x+3 y+z &=a^{2} \\ 2 x+5 y+2 a z &=0 \\ x+y+a^{2} z &=-9 \end{aligned}\right. $$ For which values of a will the system be inconsistent?

Solve each equation for x. $$ \left|\begin{array}{rr} {-2} & {x} \\ {4} & {6} \end{array}\right|=32 $$

if I is the multiplicative identity matrix of order \(2,\) find \((I-A)^{-1}\) for the given matrix \(A\) $$\left[\begin{array}{rr} 8 & -5 \\ -3 & 2 \end{array}\right]$$

Let $$ \begin{aligned} A &=\left[\begin{array}{cc} {1} & {0} \\ {0} & {1} \end{array}\right], \quad B=\left[\begin{array}{rr} {1} & {0} \\ {0} & {-1} \end{array}\right], \quad C=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {1} \end{array}\right] \\ D &=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {-1} \end{array}\right] \end{aligned} $$ Find the product of the difference between A and B and the sum of C and D.

Solve each equation for x. $$ \left|\begin{array}{cc} {x+3} & {-6} \\ {x-2} & {-4} \end{array}\right|=28 $$

If \(I\) is the multiplicative identity matrix of order \(2,\) find \((I-A)^{-1}\) for the given matrix \(A .\) $$ \left[\begin{array}{rr} {7} & {-5} \\ {-4} & {3} \end{array}\right] $$

Let $$ \begin{aligned} A &=\left[\begin{array}{cc} {1} & {0} \\ {0} & {1} \end{array}\right], \quad B=\left[\begin{array}{rr} {1} & {0} \\ {0} & {-1} \end{array}\right], \quad C=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {1} \end{array}\right] \\ D &=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {-1} \end{array}\right] \end{aligned} $$ Use any three of the matrices to verify a distributive property.

Write a system of linear equations in three or four variables to solve. Then use matrices to solve the system. Three foods have the following nutritional content per ounce. $$ \begin{array}{lccc} {} & {} & {\text { Protein }} & {\text { Vitamin } \mathrm{C}} \\ & {\text { Calories }} & {\text { (in grams) }} & {\text { (in milligrams) }} \\\ \hline \text { Food } A & {40} & {5} & {30} \\ {\text { Food } B} & {200} & {2} & {10} \\ {\text { Food } C} & {400} & {4} & {300} \end{array} $$ If a meal consisting of the three foods allows exactly 660 calories, 25 grams of protein, and 425 milligrams of vitamin \(\mathrm{C},\) how many ounces of each kind of food should be used?

You are choosing between two cellphone plans. Data Plan A offers a flat monthly rate of \(\$ 20\) per gigabyte (GB). Data Plan B has a monthly fee of \(\$ 40\) with a charge of \(\$ 15\) per GB. For how many G.B of data will the costs for the two data plans be the same? What will be the cost for each plan? (Section 1.3 Example \(3)\)

Solve each equation for x. $$ \left|\begin{array}{rrr} {1} & {x} & {-2} \\ {3} & {1} & {1} \\ {0} & {-2} & {2} \end{array}\right|=-8 $$

Find \((A B)^{-1}, A^{-1} B^{-1},\) and \(B^{-1} A^{-1} .\) What do you observe? $$ A=\left[\begin{array}{ll} {2} & {1} \\ {3} & {1} \end{array}\right] \quad B=\left[\begin{array}{ll} {4} & {7} \\ {1} & {2} \end{array}\right] $$

Let $$ \begin{aligned} A &=\left[\begin{array}{cc} {1} & {0} \\ {0} & {1} \end{array}\right], \quad B=\left[\begin{array}{rr} {1} & {0} \\ {0} & {-1} \end{array}\right], \quad C=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {1} \end{array}\right] \\ D &=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {-1} \end{array}\right] \end{aligned} $$ Use any three of the matrices to verify an associative property.

Write a system of linear equations in three or four variables to solve. Then use matrices to solve the system. A furniture company produces three types of desks: a children's model, an office model, and a deluxe model. Each desk is manufactured in three stages: cutting, construction, and finishing. The time requirements for each model and manufacturing stage are given in the following table. $$ \begin{array}{ccc} {} & {\text { Children's }} & {\text { Office }} & {\text { Deluxe }} \\\& {\text { Model }} & {\text { Model }} & {\text { Model }} \\ {\text { Cutting }} & {2 \text { hr }} & {3 \text { hr }} & {2 \text { hr }} \\\ {\text { Construction }} & {2 \text { hr }} & {1 \text { hr }} & {3 \text { hr }} \\ {\text { Finishing }} & {1 \text { hr }} & {1 \text { hr }} & {2 \text { hr }} \end{array} $$ Each week the company has available a maximum of 100 hours for cutting, 100 hours for construction, and 65 hours for finishing. If all available time must be used, how many of each type of desk should be produced each week?

Find the inverse of \(f(x)=3 x-4\).

Solve each equation for x. $$ \left|\begin{array}{rrr} {2} & {x} & {1} \\ {-3} & {1} & {0} \\ {2} & {1} & {4} \end{array}\right|=39 $$

Find \((A B)^{-1}, A^{-1} B^{-1},\) and \(B^{-1} A^{-1} .\) What do you observe? $$ A=\left[\begin{array}{rr} {2} & {-9} \\ {1} & {-4} \end{array}\right] \quad B=\left[\begin{array}{ll} {9} & {5} \\ {7} & {4} \end{array}\right] $$

Write a system of linear equations in three or four variables to solve. Then use matrices to solve the system. Imagine the entire global population as a village of precisely 200 people. The bar graph shows some numeric observations based on this scenario. (graph can't copy) Combined, there are 183 Asians, Africans, Europeans, and Americans in the village. The number of Asians exceeds the number of Africans and Europeans by \(70 .\) The difference between the number of Europeans and Americans is \(15 .\) If the number of Africans is doubled, their population exceeds the number of Europeans and Americans by \(23 .\) Determine the number of Asians, Africans, Europeans, and Americans in the global village.

A chemist needs to mix a \(75 \%\) saltwater solution with a \(50 \%\) saltwater solution to obtain 10 gallons of a \(60 \%\) saltwater solution. How many gallons of each of the solutions must be used? (Section 8.1, Example 8)

Determinants are used to find the area of a triangle whose vertices are given by three points in a rectangular coordinate system. The area of a triangle with vertices \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right)\) is $$ \text { Area }=\pm \frac{1}{2}\left|\begin{array}{lll} {x_{1}} & {y_{1}} & {1} \\ {x_{2}} & {y_{2}} & {1} \\ {x_{3}} & {y_{3}} & {1} \end{array}\right| $$ where the \(\pm\) symbol indicates that the appropriate sign should be chosen to yield a positive area. Use this information to work Exercises \(49-50\). Use determinants to find the area of the triangle whose vertices are \((3,-5),(2,6),\) and \((-3,5)\)

Prove the following statement: If \(A=\left[\begin{array}{lll}{a} & {0} & {0} \\ {0} & {b} & {0} \\ {0} & {0} & {c}\end{array}\right], a \neq 0, b \neq 0, c \neq 0\) then \(A^{-1}=\left[\begin{array}{lll}{\frac{1}{a}} & {0} & {0} \\ {0} & {\frac{1}{b}} & {0} \\ {0} & {0} & {\frac{1}{c}}\end{array}\right]\)

Solve: $$ \cos x \tan ^{2} x=3 \cos x, \quad 0 \leq x<2 \pi $$.

Determinants are used to find the area of a triangle whose vertices are given by three points in a rectangular coordinate system. The area of a triangle with vertices \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right)\) is $$ \text { Area }=\pm \frac{1}{2}\left|\begin{array}{lll} {x_{1}} & {y_{1}} & {1} \\ {x_{2}} & {y_{2}} & {1} \\ {x_{3}} & {y_{3}} & {1} \end{array}\right| $$ where the \(\pm\) symbol indicates that the appropriate sign should be chosen to yield a positive area. Use this information to work Exercises \(49-50\). Use determinants to find the area of the triangle whose vertices are \((1,1),(-2,-3),\) and \((11,-3)\)

Prove the following statement: If \(A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]\) and \(a d-b c \neq 0\) then \(A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}{d} & {-b} \\ {-c} & {a}\end{array}\right]\) (Hint: Use the method of Example 2 on page 943 .)

Explaining the Concepts What is a matrix?

In each exercise, perform the indicated operation or operations. $$ -6-(-5) $$

Determinants are used to show that three points lie on the same line (are collinear). If $$ \left|\begin{array}{lll} {x_{1}} & {y_{1}} & {1} \\ {x_{2}} & {y_{2}} & {1} \\ {x_{3}} & {y_{3}} & {1} \end{array}\right|=0 $$ then the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) are collinear. If the determinant does not equal \(0,\) then the points are not collinear. Use this information to work Exercises \(51-52\) Are the points \((3,-1),(0,-3),\) and \((12,5)\) collinear?

Explaining the Concepts Describe what is meant by the augmented matrix of a system of linear equations.

In each exercise, perform the indicated operation or operations. $$ 1(-4)+2(5)+3(-6) $$

Determinants are used to show that three points lie on the same line (are collinear). If $$ \left|\begin{array}{lll} {x_{1}} & {y_{1}} & {1} \\ {x_{2}} & {y_{2}} & {1} \\ {x_{3}} & {y_{3}} & {1} \end{array}\right|=0 $$ then the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) are collinear. If the determinant does not equal \(0,\) then the points are not collinear. Use this information to work Exercises \(51-52\) Are the points \((-4,-6),(1,0),\) and \((11,12)\) collinear?

The figure shows the letter \(L\) in a rectangular coordinate system. The figure can be represented by the matrix $$ B=\left[\begin{array}{llllll} {0} & {3} & {3} & {1} & {1} & {0} \\ {0} & {0} & {1} & {1} & {5} & {5} \end{array}\right] $$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The L is completed by connecting the last point in the matrix, (0,5), to the starting point, (0,0) . Use these ideas to solve Exercises 53-60. Use matrix operations to move the L 2 units to the left and 3 units down. Then graph the letter and its transformation in a rectangular coordinate system.

Explaining the Concepts In your own words, describe each of the three matrix row operations. Give an example with each of the operations.

In each exercise, perform the indicated operation or operations. $$ \frac{1}{2}[8-(-8)] $$

Determinants are used to write an equation of a line passing through two points. An equation of the line passing through the distinct points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) is given by $$ \left|\begin{array}{lll} {x} & {y} & {1} \\ {x_{1}} & {y_{1}} & {1} \\ {x_{2}} & {y_{2}} & {1} \end{array}\right|=0 $$ Use this information to work Exercises \(53-54\) Use the determinant to write an equation of the line passing through \((3,-5)\) and \((-2,6) .\) Then expand the determinant, expressing the line's equation in slope-intercept form.

Use the coding matrix $$ \begin{aligned} &A=\left[\begin{array}{rrr} {1} & {-1} & {0} \\ {3} & {0} & {2} \\ {-1} & {0} & {-1} \end{array}\right] \text { and its inverse }\\\ &A^{-1}=\left[\begin{array}{rrr} {0} & {1} & {2} \\ {-1} & {1} & {2} \\ {0} & {-1} & {-3} \end{array}\right] \text { to write a cryptogram for each } \end{aligned} $$ message. Check your result by decoding the cryptogram. \(\begin{array}{llll}{-\mathrm{S}} & {\mathrm{E}} & {\mathrm{N}} & {\mathrm{D}}\end{array}\) \(\mathbf{A} \quad \mathbf{S} \quad \mathbf{H}\) \(\begin{array}{lllllllll}{19} & {5} & {14} & {4} & {0} & {3} & {1} & {19} & {8}\end{array}\) $$ \text { Use }\left[\begin{array}{ccc} {19} & {4} & {1} \\ {5} & {0} & {19} \\ {14} & {3} & {8} \end{array}\right] $$

The figure shows the letter \(L\) in a rectangular coordinate system. The figure can be represented by the matrix $$ B=\left[\begin{array}{llllll} {0} & {3} & {3} & {1} & {1} & {0} \\ {0} & {0} & {1} & {1} & {5} & {5} \end{array}\right] $$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The L is completed by connecting the last point in the matrix, (0,5), to the starting point, (0,0) . Use these ideas to solve Exercises 53-60. Use matrix operations to move the L 2 units to the right and 3 units down. Then graph the letter and its transformation in a rectangular coordinate system.

Explaining the Concepts Describe how to use row operations and matrices to solve a system of linear equations.

Determinants are used to write an equation of a line passing through two points. An equation of the line passing through the distinct points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) is given by $$ \left|\begin{array}{lll} {x} & {y} & {1} \\ {x_{1}} & {y_{1}} & {1} \\ {x_{2}} & {y_{2}} & {1} \end{array}\right|=0 $$ Use this information to work Exercises \(53-54\) Use the determinant to write an equation of the line passing through \((-1,3)\) and \((2,4) .\) Then expand the determinant, expressing the line's equation in slope-intercept form.

Explaining the Concepts What is the difference between Gaussian elimination and Gauss-Jordan elimination?

Explain how to evaluate a second-order determinant.

What is the multiplicative identity matrix?

Describe the determinants \(D_{x}\) and \(D_{y}\) in terms of the coefficients and constants in a system of two equations in two variables.

If you are given two matrices, \(A\) and \(B,\) explain how to determine if \(B\) is the multiplicative inverse of \(A\).

The figure shows the letter \(L\) in a rectangular coordinate system. The figure can be represented by the matrix $$ B=\left[\begin{array}{llllll} {0} & {3} & {3} & {1} & {1} & {0} \\ {0} & {0} & {1} & {1} & {5} & {5} \end{array}\right] $$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The L is completed by connecting the last point in the matrix, (0,5), to the starting point, (0,0) . Use these ideas to solve Exercises 53-60. a. If \(A=\left[\begin{array}{rr}{1} & {0} \\ {0} & {-1}\end{array}\right],\) find \(A B\). b. Graph the object represented by matrix \(A B .\) What effect does the matrix multiplication have on the letter \(L\) represented by matrix \(B ?\)

Explain how to evaluate a third-order determinant.

Explain why a matrix that does not have the same number of rows and columns cannot have a multiplicative inverse.

The figure shows the letter \(L\) in a rectangular coordinate system. The figure can be represented by the matrix $$ B=\left[\begin{array}{llllll} {0} & {3} & {3} & {1} & {1} & {0} \\ {0} & {0} & {1} & {1} & {5} & {5} \end{array}\right] $$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The L is completed by connecting the last point in the matrix, (0,5), to the starting point, (0,0) . Use these ideas to solve Exercises 53-60. a. If \(A=\left[\begin{array}{rr}{-1} & {0} \\ {0} & {1}\end{array}\right],\) find \(A B\). b. Graph the object represented by matrix \(A B .\) What effect does the matrix multiplication have on the letter \(\mathrm{L}\) represented by matrix \(B ?\)

When expanding a determinant by minors, when is it necessary to supply minus signs?

Explain how to find the multiplicative inverse for a \(2 \times 2\) invertible matrix.