Review Exercises

Concepts and Vocabulary

In Exercises 1–4, determine the order of each matrix.

  1. [−1234][−1234]

  2. [5]

  3. [23−1201]⎡⎣⎢2−10321⎤⎦⎥

  4. [1−12−45432][15−1423−42]

  5. Let A be the matrix of Exercise 4. Identify the entries a12,  a14,  a23,  and a21.a12,  a14,  a23,  and a21.

  6. Write the 2×32×3 matrix A with entries a22=3,  a21=4a22=3,  a21=4, a12=2,  a13=5,  a11=−1,  and a23=1.a12=2,  a13=5,  a11=−1,  and a23=1.

In Exercises 7 and 8, write the augmented matrix for each system of linear equations.

  1. {2x−3y=73x+y=6{2x3x−+3yy==76

  2. {x+y−z=62x−3y−2z=25x−3y+z=8⎧⎩⎨⎪⎪x2x5x+−−y3y3y−−+z2zz===628

In Exercises 9 and 10, convert each matrix to row-echelon form (Answers may vary).

  1. [012120341−217]⎡⎣⎢02110−2231147⎤⎦⎥

  2. [3−121211−121217]⎡⎣⎢311−1122−111227⎤⎦⎥

In Exercises 11 and 12, convert each matrix to reduced row-echelon form.

  1. [313121111−1−10]⎡⎣⎢32111−131−1110⎤⎦⎥

  2. [34−4221−211121]⎡⎣⎢321411−4−22211⎤⎦⎥

In Exercises 13–16, solve each system of equations by Gaussian elimination.

  1. {x+y−z=02x+y−2z=33x−2y+3z=9⎧⎩⎨⎪⎪x2x3x++−yy2y−−+z2z3z===039

  2. {x−y+2z=1x+3y−z=62x+y−3z=1⎧⎩⎨⎪⎪xx2x−++y3yy+−−2zz3z===161

  3. {x−2y+3z=−22x−3y+z=93x−y+2z=5⎧⎩⎨⎪⎪x2x3x−−−2y3yy+++3zz2z===−295

  4. {2x+y+z=7x+2y+z=3x+2y+2z=6⎧⎩⎨⎪⎪2xxx+++y2y2y+++zz2z===736

In Exercises 17–20, solve each system of equations by Gauss–Jordan elimination.

  1. {x−2y−2z=113x+4y−z=−24x+5y+7z=7⎧⎩⎨⎪⎪x3x4x−++2y4y5y−−+2zz7z===11−27

  2. {x−y−9z=13x+2y−z=24x+3y+3z=0⎧⎩⎨⎪⎪x3x4x−++y2y3y−−+9zz3z===120

  3. {2x−y+3z=4x+3y+3z=−23x+2y−6z=6⎧⎩⎨⎪⎪2xx3x−++y3y2y++−3z3z6z===4−26

  4. {x−y−9z=13x+2y−z=24x+3y+3z=0⎧⎩⎨⎪⎪x3x4x−++y2y3y−−+9zz3z===120

  5. Find x and y if [x−y01x+y]=[1013].[x−1yx0+y]=[1103].

  6. Find x and y if [2x+3y−1−201x−y23x+4y5]=[4−1−201−3255].⎡⎣⎢2x+023y3x−11+4yx−2−5y⎤⎦⎥=⎡⎣⎢402−115−2−35⎤⎦⎥.

  7. Let A=[12−34]A=[1−324] and B=[2−3−56].B=[2−5−36]. Find each of the following.

    1. A+BA+B

    2. A−BA−B

    3. 2A

    4. −3B−3B

    5. 2A−3B2A−3B

  8. Let A=[20−1122−203]A=⎡⎣⎢21−2020−123⎤⎦⎥ and B=[01−11−10−101].B=⎡⎣⎢01−11−10−101⎤⎦⎥.

    Find each of the following.

    1. A+BA+B

    2. A−BA−B

    3. 2A+3B2A+3B

  9. For the matrices A and B of Exercise 23, solve the matrix equation 3A+2B−3X=03A+2B−3X=0 for X.

  10. For the matrices A and B of Exercise 24, solve the matrix equation A−2X+2B=0A−2X+2B=0 for X.

In Exercises 27–30, find the products, if possible.

  1. AB

  2. BA

  1. A=[0123], B=[−1−1−34]A=[0213], B=[−1−3−14]

  2. A=[012−101], B=[123−104]A=[0−11021], B=⎡⎣⎢1302−14⎤⎦⎥

  3. A=[12−1], B=[231]A=[12−1], B=⎡⎣⎢231⎤⎦⎥

  4. A=[102−13−2], B=[12342−123]A=⎡⎣⎢1230−1−2⎤⎦⎥, B=[122−13243]

  5. Find a matrix A=[xyzw]A=[xzyw] such that

    [123−1]A=[561−3].
    [132−1]A=[516−3].

  6. Find a matrix B=[xyzw]B=[xzyw] such that

    B[123−1]=[561−3].
    B[132−1]=[516−3].

    Compare your answers for Exercise 31 and 32.

  7. Find a matrix A=[xyzw]A=[xzyw] such that

    [5342]A=[1001].
    [5432]A=[1001].

  8. Find a matrix A=[xyzw]A=[xzyw] such that

    A[5342]=[1001].
    A[5432]=[1001].

    Compare your answers for Exercise 33 and 34.

  9. Let A=[01−13].A=[0−113].

    1. Find A−1.A−1. (Use the formula on page 626.)

    2. Find (A2)−1.(A2)−1.

    3. Find (A−1)2.(A−1)2.

    4. Use your answers in parts (b) and (c) to show that (A−1)2(A−1)2 is the inverse of A2.A2.

  10. Repeat Exercise 35 for A=[3423].A=[3243].

In Exercises 37–40, show that B is the inverse of A.

  1. A=[7665],B=[−566−7]A=[7665],B=[−566−7]

  2. A=[−131−4],B=[−4−3−1−1]A=[−113−4],B=[−4−1−3−1]

  3. A=[2−101041−11],B=15[41−432−8−111]A=⎡⎣⎢211−10−1041⎤⎦⎥,B=15⎡⎣⎢43−1121−4−81⎤⎦⎥

  4. A=[2010−12101],B=[10−1−2−14−102]A=⎡⎣⎢2010−10121⎤⎦⎥,B=⎡⎣⎢1−2−10−10−142⎤⎦⎥

In Exercises 41–44, find the inverse (if it exists) of each matrix.

  1. [3124][3214]

  2. [3−412][31−42]

  3. [12−2−1300−21]⎡⎣⎢1−1023−2−201⎤⎦⎥

  4. [121224003]⎡⎣⎢120220143⎤⎦⎥

In Exercises 45–50, use an inverse matrix (if possible) to solve each system of linear equations.

  1. {x+3y=72x+5y=4{x2x++3y5y==74

  2. {3x+5y=42x+4y=5{3x2x++5y4y==45

  3. {x+3y+3z=3x+4y+3z=5x+3y+4z=6⎧⎩⎨⎪⎪xxx+++3y4y3y+++3z3z4z===356

  4. {x+2y+3z=62x+4y+5z=83x+5y+6z=10⎧⎩⎨⎪⎪x2x3x+++2y4y5y+++3z5z6z===6810

  5. {x−y+z=34x+2y=57x−y−z=6⎧⎩⎨⎪⎪x4x7x−+−y2yy+−zz===356

  6. {x+2y+4z=74x+3y−2z=6x−3z=4⎧⎩⎨⎪⎪x4xx++2y3y+−−4z2z3z===764

In Exercises 51–54, find the determinant of each matrix.

  1. [2−534][23−54]

  2. [−1411−3][−1114−3]

  3. [122147][124217]

  4. [−79−45][−7−495]

In Exercises 55 and 56, find (a) the minors M12, M23,M12, M23, and M22M22 and (b) the cofactors A12, A23,A12, A23, and A22A22 of the given matrix A.

  1. A=[4−12−2−3502−4]A=⎡⎣⎢4−20−1−3225−4⎤⎦⎥

  2. A=[123456789]A=⎡⎣⎢147258369⎤⎦⎥

In Exercises 57 and 58, find the determinant of each matrix A by expanding (a) by the second row and (b) by the third column.

  1. A=[1−234−1−2−215]A=⎡⎣⎢14−2−2−113−25⎤⎦⎥

  2. A=[1436810254]A=⎡⎣⎢1624853104⎤⎦⎥

In Exercises 59 and 60, find the determinant of each matrix A.

  1. A=[120012102]A=⎡⎣⎢101210022⎤⎦⎥

  2. A=[2342592716]A=⎡⎣⎢2223574916⎤⎦⎥

In Exercises 61–64, use Cramer’s Rule to solve each system of equations.

  1. {5x+3y=112x+y=4{5x2x++3yy==114

  2. {2x−7y−13=05x+6y−9=0{2x5x−+7y6y−−139==00

  3. {3x+y−z=14x+3y−z=16x+y−3z=−10⎧⎩⎨⎪⎪3xxx+++y3yy−−−zz3z===1416−10

  4. {2x+3y−2z=05y−3x+4z=93x+7y−6z=4⎧⎩⎨⎪⎪2x5y3x+−+3y3x7y−+−2z4z6z===094

In Exercises 65 and 66, solve each equation for x.

  1. |124−3571x4|=0∣∣∣∣1−3125x474∣∣∣∣=0

  2. |1−120x132x−1|=14∣∣∣∣103−1x221x−1∣∣∣∣=14

  3. Show that the equation of the line passing through the points (2, 3) and (1, −4)(1, −4) is equivalent to the equation

    |xy12311−41|=0. (see page 646.)
    ∣∣∣∣x21y3−4111∣∣∣∣=0. (see page 646.)

  4. Use a determinant to find the area of the triangle with vertices A(1, 1), B(4, 3), and C(2, 7). (See page 645.)

Applying the Concepts

  1. Maximizing profit. A company is considering which of three methods of production it should use in producing the three products A, B, and C. The number of units of each product produced by each method is shown in this matrix:

    ABCMethod 1Method 2Method 3[482571548]
    Method 1Method 2Method 3ABC⎡⎣⎢455874218⎤⎦⎥

    The profit per unit of the products A, B, and C is $10, $4, and $6, respectively. Use matrix multiplication to find which method maximizes total profit for the company.

  2. Nutrition. Consider two families: Ardestanis (A) and Barkley (B). Family A consists of two adult males, three adult females, and one child. Family B consists of one adult male, one adult female, and two children. Their mutual dietician considers the age and weight of each member of the two families and recommends daily allowances for calories—adult males 2200, adult females 1700, and children 1500—and for protein—adult males 50 grams, adult females 40 grams, and children 30 grams.

    Represent the information in the preceding paragraph by matrices. Use matrix multiplication to calculate the total requirements of calories and proteins for each family.

In Exercises 71–76, each problem produces a system of linear equations in two or three variables. Use the method of your choice from this chapter to solve the system of equations.

  1. Airplane speed. An airplane traveled 1680 miles in 3 hours with the wind. It would have taken 3.5 hours to make the same trip against the wind. Find the speed of the plane and the velocity of the wind assuming that both are constant.

  2. Walking speed. In 5 hours, Nertha walks 2 miles farther than Kristina walks in 4 hours; then in 12 hours, Kristina walks 10 miles farther than Nertha walks in 10 hours. How many miles per hour does each woman walk?

  3. Friends. Andrew, Bonnie, and Chauncie have $320 between them. Andrew has twice as much as Chauncie, and Bonnie and Chauncie together have $20 less than Andrew. How much money does each person have?

  4. Ratio. Steve and Nicole entered a double-decker bus in London on which there were 5858 as many men as women. After they boarded the bus, there were 711711 as many men as women on the bus. How many people were on the bus before they entered it?

  5. Hospital workers. Metropolitan Hospital employs three types of caregivers: registered nurses, licensed practical nurses, and nurse’s aides. Each registered nurse is paid $75 per hour, each licensed practical nurse is paid $20 per hour, and each nurse’s aide is paid $30 per hour. The hospital has budgeted $4850 per hour for the three categories of caregivers. The hospital requires that each caregiver spend some time in a class learning about advanced medical practices each week: registered nurses, 3 hours; licensed practical nurses, 5 hours; and nurse’s aides, 4 hours. The advanced classes can accommodate a maximum of 530 person-hours each week. The hospital needs a total of 130 registered nurses, licensed practical nurses, and nurse’s aides to meet the daily needs of the patients. Assume that the allowable payroll is met, the training classes are full, and the patients’ needs are met. How many registered nurses, licensed practical nurses, and nurse’s aides should the hospital employ?

  6. Sales tax. A drugstore sells three categories of items:

    1. Medicine, which is not taxed

    2. Nonmedical items, which are taxed at the rate of 8%

    3. Beer and cigarettes, which are taxed at the “sin tax” rate of 20%

    Last Friday, the total sales (excluding taxes) of all three items were $18,500. The total tax collected was $1020. The sales of medicines exceeded the combined sales of the other items by $3500. Find the sales in each category.

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset
3.144.202.167