Ch2la

chapter 2 Linear Algebra exam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Perform the matrix operation.

1) Let A = -3 1 . Find 5A. 02 A) -15 5 02

B)

C)

-15 5 0 10

-15 1 02

D)

26 57

Answer: B 2) Let B = -1 3 4 -3 . Find -2B. A) 2 -6 -8 6 B)

C)

-3 1 2 -5

2 3 4 -3

D)

-2 6 8 -6

Answer: A 4 3) Let C = -2 8 A) 2 -1 4

. Find (1/2) C.

B)

C)

4 -1 8

2 -2 8

D)

8 -4 16

Answer: A 4) Let A = 3 3 25 A) 9 13 1 11

and B =

04 -1 6

. Find 3A + B.

B)

C)

9 13 5 21

9 7 5 11

D)

9 21 3 33

Answer: B 1 5) Let C = -3 2 A) -3 9 -6

and D =

-1 3 -2

. Find C - 4D.

B)

C) -5 15 -10

5 -6 4

D)

5 -15 10

Answer: D 6) Let A = -5 2 A) -7 4

and B = 1 0

. Find 2A + 3B. B) -9 4

C) -2 2

Answer: A

1

D) -10 4

7) Let A = A)

-5 -2 7 -6 5 8

-6 -8 and B = -2 5 . Find A + B. 6 -2 B) -11 -6 5 -1 11 6

-11 -10 -5 -6 11 -6

C)

16 99 -1 5

D) -11 -10 5 -1 11 6

Answer: D 8) Let A = -5 3 and B = 5 4 . Find A - B. -2 -10 -2 9 A) B) 0 1 10 -1 -4 19 -4 -1

C)

0 -1 0 -19

D) -10 -1 0 -19

Answer: D 9) Let A = -10 5 and B = 0 0 . Find A + B. 3 -10 00 A) B) 00 -10 5 3 -10 00

C)

10 -5 -3 10

D) Undefined

Answer: A Find the matrix product AB, if it is defined.

10) A = -1 3 , B = 0 -2 5 .

36 1 -3 2 A)

3 6 -7 -24 1 27

B)

C) AB is undefined.

D)

B) -8 -18 0 -6

C) -6 0 30 -8

D) -18 -8 -6 0

B) AB is undefined.

C)

D)

0 -6 15 3 -18 12

3 -7 1 6 -24 27

Answer: D 11) A = 3 -2 3 0 A)

, B = 0 -2 .

4 6

04 12 0

Answer: B 12) A = 1 3 -3 3 0 5 A)

30 , B = -3 1 .

05

-6 -12 9 25

-12 -6 25 9

Answer: A

2

3 -9 0 0 0 25

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, if the products are defined.

13) A is 2 × 2, B is 2 × 2.

A) AB is 2 × 4, BA is 2 × 4.

B) AB is 1 × 1, BA is 1 × 1.

C) AB is 4 × 2, BA is 4 × 2.

D) AB is 2 × 2, BA is 2 × 2.

Answer: D 14) A is 2 × 1, B is 1 × 1.

A) AB is 1 × 2, BA is 1 × 1.

C) AB is 2 × 2, BA is 1 × 1.

B) AB is undefined, BA is 1× 2.

D) AB is 2 × 1, BA is undefined.

Answer: D Find the transpose of the matrix.

8 2 15) -2 0 -7 7 A) -7 7 -2 0 8 2

B)

C)

2 8 0 -2 7 -7

2 0 7 8 -2 -7

D)

8 -2 -7 2 0 7

Answer: D 16) 8 2 8 2 0 -3 0 -3 A) 0 -3 0 -3 8 2 8 2

B)

C)

8 0 2 -3 8 0 2 -3

0 -3 0 -3

Answer: B Decide whether or not the matrices are inverses of each other.

17) 5 3 and 2 -3 32 -3 5 A) Yes

B) No

Answer: A 18) 10 1 and 0 1 -1 0 -1 10 A) No

B) Yes

Answer: A

19) -2 4 4 -4

and

1 1 2 4 1 1 2 4

A) Yes

B) No

Answer: B

3

8 2 8 2

D)

2 8 2 8 -3 0 -3 0

20) -5 1 -7 1

1 1 2 - 2

and

7 5 2 - 2

A) Yes

B) No

Answer: A

21)

6 -5 -3 5

and

1 1 3 3 1 2 5 5

A) Yes

B) No

Answer: A 0.2 22) 9 4 and -0.2 44 0.2 -0.45 A) No

B) Yes

Answer: A

23) 9 -2 7 -2

0.5 0.5 and - 7 - 9 4 4

A) Yes

B) No

Answer: B

24) -5 -1 6 0

and

0

1 6

-1

5 6

A) Yes

B) No

Answer: B 2 -1 0 25) -1 1 -2 1 0 -1 A) No

1 -1 2 and -3 -2 4 -1 1 1

B) Yes

Answer: A Find the inverse of the matrix, if it exists.

26) A = -3 3 -5 -2 A) B) 2 1 1 1 21 7 7 7 5 1 21 7

C)

5 2 21 21

5 1 21 7 -

Answer: A

4

2 1 21 7

D) -

2 21

1 7

-

5 1 21 7

27) A = 0 1 3 -6 A) 1 0 1 2 3

B) 2

C)

1 3

0

1 0

D)

1 3

2 -

1 2

1 3

-1

0

Answer: B 28) A = A)

6 0 -2 -1

B) A is not invertible

1 6 -

C)

0

1 -1 3

D)

1 6

0

1 -1 3

-1 0 1 1 3 6

Answer: A 29) A = A)

1 -2 -2 4

B)

1 1 3 6 2 1 3 3

C)

2 1 3 3 -

1 3

1 6

D) A is not invertible

2 1 3 3 1 1 3 6

Answer: D 30) A = -5 4 0 4 A)

1 0 4 -

B)

1 1 5 5

C)

1 1 5 5 0

1 4

1 4

1 5

0-

1 5

D) -

1 1 5 5 0

1 4

Answer: D 31) A = 1 3 20 A) 1 1 3 6 0

1 2

B)

C)

1 1 6 2 1 0 3

1 0 2 -

Answer: D

5

1 1 3 6

D) 0

1 2

1 1 3 6

32)

100 -1 1 0 111 A)

1 0 0 1 1 0 -2 -1 1

B)

C)

111 011 001

-1 0 0 -1 -1 0 -1 -1 -1

D)

1 -1 1 0 1 -1 0 0 1

Answer: A Solve the system by using the inverse of the coefficient matrix.

33) 3x + 4y = 12 2x + 2y = 4 A) (-4, -6) B) (6, -4)

C) No solution

D) (-4, 6)

B) No solution

C) (2, -3)

D) (-3, 2)

B) (-2, -2)

C) (7, 2)

D) 1 -

B) (1, -2)

C) (2, -1)

D) (-1, 2)

B) (-2, -3)

C) (3, 2)

D) (-3, -2)

B) (-1, -4)

C) (1, 4)

D) (-4, -1)

B) (2, 5)

C) (-2, 5)

D) (5, 2)

Answer: D 34) 3x + 4y = -1 3x = -9 A) (-3, -2) Answer: D 35) 7x - 4y = 7 -14x + 8y = 2 A) No solution

7 y, y 4

Answer: A 36) -2x - 6y = -2 2x - y = -5 A) (-2, 1) Answer: A 37) -3x + 9y = 9 3x + 2y = 13 A) (2, 3) Answer: C 38) -10x + 4y = 6 6x - y = 2 A) (4, 1) Answer: C 39) -2x + 4y = 2 3x + 4y = -23 A) (-5, -2) Answer: A

6

40) -5x + 3y = 8 -2x + 4y = 20 A) (6, 2)

B) (-6, -2)

C) (2, 6)

D) (-2, -6)

C) A-1 does not exist.

1 0 1 D) A-1 = 0 1 -1 0 0 0

Answer: C Find the inverse of the matrix A, if it exists.

3 -4 7 41) A = 3 -3 6 6 -7 13 1 01 3 3 6 A) A-1 = 0 1 -1 B) A-1 = -4 -3 -7 0 10 7 6 13 Answer: C 111 42) A = 2 1 1 223 -1 -1 -1 B) A-1 = -2 -1 -1 -2 -2 -3

A) A-1 does not exist.

C) A-1 =

1 1 1 1 2 1 2

1 1

D) A-1 =

1 1 2 3

-1 1 0 4 -1 -1 -2 0 1

Answer: D 1 3 2 43) A = 1 3 3 2 7 8

-1 -3 -2 A) A-1 = -1 -3 -3 -2 -7 -8

C) A-1 =

B) A-1 does not exist.

-3 10 -3 2 -4 1 -1 1 0

Answer: C

7

1

1 3

1 2

D) A-1 = 1

1 3

1 3

1 2

1 7

1 8

108 44) A = 1 2 3 253

9 A) A-1 = -3 -1 11 C) A-1 = 0 2 83

-40 16 13 -5 5 -2 2 5 3

-1 0 -8 B) A-1 = -1 -2 -3 -2 -5 -3

D) A-1 does not exist.

Answer: A 4 -2 3 45) A = 9 -3 6 5 -1 3

1 3

2 9

5 B) A-1 = 9

A) A-1 does not exist.

2 3

-

4 3

2 3

4 -3 5

1 3

1 1 4 9

4 9 5 C) A-1 = -2 -3 -1 3 6 3

1 3

1 1 D) A-1 = 9 - 3

1 6

1 -1 5

1 3

Answer: A 0 3 3 46) A = -5 0 4 0 8 0 4 15

A) A-1 =

C) A-1 =

1 5

0

0

1 3

0

-

1 10

-

1 8 -

0

1 3

1 5

0

0

1 10

1 8

-

B) A-1 =

1 8

4 15 -

-

4 15

-

-

1 5

Answer: A

8

1 10

1 8

0

1 8

1 3

0

0

D) A-1 does not exist.

1 8

-

Determine whether the matrix is invertible.

47) 6 1 3 4 A) Yes

B) No

Answer: A 48)

8 5 -8 7 2 -7 -4 0 4 A) No

B) Yes

Answer: A

9