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