Q. If A = 6i + 8j and B = 3i + 4j, what is the scalar product A · B?
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Solution
A · B = (6)(3) + (8)(4) = 18 + 32 = 50.
Correct Answer:
B
— 54
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Q. If A = 6i + 8j and B = 3i + 4j, what is the value of A · B?
Show solution
Solution
A · B = (6)(3) + (8)(4) = 18 + 32 = 50.
Correct Answer:
C
— 48
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Q. If A = 6i - 2j and B = -3i + 4j, find A · B.
A.
-18
B.
-24
C.
-12
D.
-6
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Solution
A · B = (6)(-3) + (-2)(4) = -18 - 8 = -26.
Correct Answer:
A
— -18
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Q. If A = 7i + 1j and B = 1i + 7j, calculate A · B.
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Solution
A · B = (7)(1) + (1)(7) = 7 + 7 = 14.
Correct Answer:
A
— 56
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Q. If A = 7i + 1j and B = 1i + 7j, find A · B.
Show solution
Solution
A · B = (7)(1) + (1)(7) = 7 + 7 = 14.
Correct Answer:
A
— 56
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Q. If A = 7i + 1j and B = 2i + 3j, what is the scalar product A · B? (2022)
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Solution
A · B = (7)(2) + (1)(3) = 14 + 3 = 17
Correct Answer:
A
— 23
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Q. If A = 7i + 24j and B = 24i - 7j, calculate A · B.
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Solution
A · B = (7)(24) + (24)(-7) = 168 - 168 = 0.
Correct Answer:
A
— 0
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Q. If A = 7i - 2j and B = -3i + 4j, find A · B. (2021)
A.
-26
B.
-28
C.
-30
D.
-32
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Solution
A · B = (7)(-3) + (-2)(4) = -21 - 8 = -29.
Correct Answer:
A
— -26
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Q. If A = 7i - 3j and B = -2i + 5j, what is the value of A · B?
A.
-29
B.
-31
C.
-25
D.
-27
Show solution
Solution
A · B = (7)(-2) + (-3)(5) = -14 - 15 = -29.
Correct Answer:
A
— -29
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Q. If A = ai + bj and B = ci + dj, what is the expression for A · B?
A.
ac + bd
B.
ab + cd
C.
ad + bc
D.
a + b + c + d
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Solution
A · B = (ai + bj) · (ci + dj) = ac + bd.
Correct Answer:
A
— ac + bd
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Q. If A = i + 2j + 3k and B = 4i + 5j + 6k, find A · B.
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Solution
A · B = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32.
Correct Answer:
B
— 30
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Q. If A = i + 2j + 3k and B = 4i + 5j + 6k, what is A - B? (2023)
A.
-3i - 3j - 3k
B.
-3i - 3j + 3k
C.
3i + 3j + 3k
D.
3i + 3j - 3k
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Solution
A - B = (1 - 4)i + (2 - 5)j + (3 - 6)k = -3i - 3j - 3k.
Correct Answer:
A
— -3i - 3j - 3k
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Q. If A = i + 2j + 3k and B = 4i + 5j + 6k, what is the value of A · B?
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Solution
A · B = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32.
Correct Answer:
B
— 30
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Q. If A = [[1, 0], [0, 1]] and B = [[2, 3], [4, 5]], what is AB?
A.
[2, 3], [4, 5]
B.
[1, 0], [0, 1]
C.
[0, 0], [0, 0]
D.
[6, 8], [12, 15]
Show solution
Solution
AB = [[1*2 + 0*4, 1*3 + 0*5], [0*2 + 1*4, 0*3 + 1*5]] = [[2, 3], [4, 5]].
Correct Answer:
A
— [2, 3], [4, 5]
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Q. If A = [[1, 0], [0, 1]] is the identity matrix, what is A^n for any integer n?
A.
A
B.
0
C.
I
D.
None of the above
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Solution
A^n = I for any integer n, where I is the identity matrix.
Correct Answer:
C
— I
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Q. If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], what is A + B?
A.
[6, 8], [10, 12]
B.
[1, 2], [3, 4]
C.
[5, 6], [7, 8]
D.
[8, 10], [10, 12]
Show solution
Solution
A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]].
Correct Answer:
A
— [6, 8], [10, 12]
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Q. If A = [[1, 2], [3, 4]], find A^2.
A.
[7, 10], [15, 22]
B.
[1, 2], [3, 4]
C.
[10, 13], [22, 29]
D.
[-1, -2], [-3, -4]
Show solution
Solution
A^2 = A * A = [[1*1 + 2*3, 1*2 + 2*4], [3*1 + 4*3, 3*2 + 4*4]] = [[7, 10], [15, 22]].
Correct Answer:
A
— [7, 10], [15, 22]
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Q. If A = [[1, 2], [3, 4]], find the determinant of A.
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Solution
The determinant of A is (1*4) - (2*3) = 4 - 6 = -2.
Correct Answer:
B
— 2
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Q. If A = [[1, 2], [3, 4]], what is A^2?
A.
[7, 10; 15, 22]
B.
[1, 2; 3, 4]
C.
[10, 14; 22, 30]
D.
[-1, -2; -3, -4]
Show solution
Solution
A^2 = A * A = [[1*1 + 2*3, 1*2 + 2*4], [3*1 + 4*3, 3*2 + 4*4]] = [[7, 10], [15, 22]].
Correct Answer:
A
— [7, 10; 15, 22]
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Q. If A = [[1, 2], [3, 4]], what is the adjoint of A?
A.
[[4, -2], [-3, 1]]
B.
[[1, 3], [2, 4]]
C.
[[2, 1], [4, 3]]
D.
[[0, 0], [0, 0]]
Show solution
Solution
The adjoint of A is [[4, -2], [-3, 1]].
Correct Answer:
A
— [[4, -2], [-3, 1]]
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Q. If A = [[1, 2], [3, 4]], what is the determinant of A?
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Solution
The determinant of A is (1*4) - (2*3) = 4 - 6 = -2.
Correct Answer:
A
— -2
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Q. If A = [[1, 2], [3, 4]], what is the eigenvalue of A?
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Solution
The eigenvalues are found from the characteristic polynomial λ^2 - 5λ + 2 = 0, which gives λ = 5.
Correct Answer:
A
— 5
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Q. If A = [[1, 2], [3, 4]], what is the inverse of A?
A.
[[4, -2], [-3, 1]]
B.
[[-2, 1], [1.5, -0.5]]
C.
[[-2, 1], [1.5, -0.5]]
D.
[[4, -2], [-3, 1]]
Show solution
Solution
The inverse of A is (1/det(A)) * adj(A) = (1/(-2)) * [[4, -2], [-3, 1]] = [[-2, 1], [1.5, -0.5]].
Correct Answer:
A
— [[4, -2], [-3, 1]]
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Q. If A = [[1, 2], [3, 4]], what is the trace of A?
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Solution
The trace of A is the sum of the diagonal elements: 1 + 4 = 5.
Correct Answer:
A
— 5
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Q. If A = [[2, 0], [0, 3]], what is the eigenvalue of A?
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Solution
The eigenvalues of A are the diagonal elements: 2 and 3.
Correct Answer:
B
— 3
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Q. If A = [[2, 3], [1, 4]], what is A^2? (2020)
A.
[[7, 18], [18, 7]]
B.
[[12, 21], [21, 12]]
C.
[[12, 21], [21, 16]]
D.
[[10, 21], [21, 10]]
Show solution
Solution
A^2 = A * A = [[2*2 + 3*1, 2*3 + 3*4], [1*2 + 4*1, 1*3 + 4*4]] = [[10, 21], [21, 16]].
Correct Answer:
C
— [[12, 21], [21, 16]]
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Q. If A = [[2, 3], [1, 4]], what is the inverse of A?
A.
[[4, -3], [-1, 2]]
B.
[[4, 3], [-1, 2]]
C.
[[2, -3], [-1, 2]]
D.
[[3, -2], [-1, 2]]
Show solution
Solution
The inverse of A is given by (1/det(A)) * adj(A). Det(A) = (2*4) - (3*1) = 5. The adjoint is [[4, -3], [-1, 2]]. Thus, A^(-1) = (1/5) * [[4, -3], [-1, 2]].
Correct Answer:
A
— [[4, -3], [-1, 2]]
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Q. If A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, what is A ∪ B?
A.
{1, 2, 3, 4, 6, 8}
B.
{2, 4}
C.
{1, 3, 5, 7}
D.
{1, 2, 3, 4, 5}
Show solution
Solution
The union A ∪ B includes all elements from both sets, which are {1, 2, 3, 4, 6, 8}.
Correct Answer:
A
— {1, 2, 3, 4, 6, 8}
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Q. If A = {1, 2, 3} and B = {1, 2, 3, 4, 5}, what is A ⊆ B?
A.
True
B.
False
C.
Depends on the context
D.
None of the above
Show solution
Solution
Set A is a subset of set B because all elements of A are contained in B. Therefore, A ⊆ B is True.
Correct Answer:
A
— True
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Q. If A = {1, 2, 3} and B = {1, 2, 3, 4}, what is A ⊆ B?
A.
True
B.
False
C.
Depends on A
D.
Not enough information
Show solution
Solution
A is a subset of B if all elements of A are in B. Here, all elements of A are in B, so A ⊆ B is true.
Correct Answer:
A
— True
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