Q. If A = (2, 3, 4) and B = (x, y, z) such that A · B = 20, find the value of x + y + z.
Solution
A · B = 2x + 3y + 4z = 20. If we assume x = 2, y = 2, z = 2, then 2*2 + 3*2 + 4*2 = 20, thus x + y + z = 6.
Correct Answer: C — 7
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Q. If A = (3, -1, 2) and B = (k, 4, -1) are orthogonal, find k.
Solution
A · B = 3k - 4 - 2 = 0. Thus, 3k - 6 = 0, k = 2.
Correct Answer: A — -2
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Q. If A = (3, -2, 1) and B = (4, 0, -1), what is the value of A · B?
Solution
A · B = 3*4 + (-2)*0 + 1*(-1) = 12 + 0 - 1 = 11.
Correct Answer: A — -1
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Q. If A = (a, b, c) and B = (1, 2, 3) such that A · B = 0, what is the relation between a, b, and c?
-
A.
a + 2b + 3c = 0
-
B.
a - 2b + 3c = 0
-
C.
a + b + c = 0
-
D.
a - b - c = 0
Solution
A · B = a*1 + b*2 + c*3 = 0. Thus, a + 2b + 3c = 0.
Correct Answer: A — a + 2b + 3c = 0
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Q. If A = (a, b, c) and B = (1, 2, 3), and A · B = 14, find a + b + c.
Solution
a*1 + b*2 + c*3 = 14. One possible solution is a = 2, b = 4, c = 0, so a + b + c = 6.
Correct Answer: C — 7
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Q. If A = (a, b, c) and B = (1, 2, 3), and A · B = 14, what is the equation?
-
A.
a + 2b + 3c = 14
-
B.
a - 2b + 3c = 14
-
C.
a + 2b - 3c = 14
-
D.
a - 2b - 3c = 14
Solution
A · B = a*1 + b*2 + c*3 = 14.
Correct Answer: A — a + 2b + 3c = 14
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Q. If A = (a, b, c) and B = (1, 2, 3), find the value of a if A · B = 10.
Solution
A · B = a*1 + b*2 + c*3 = 10. If b = 2 and c = 1, then a + 4 + 3 = 10, so a = 3.
Correct Answer: C — 3
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Q. If A = (a, b, c) and B = (1, 2, 3), what is the scalar product A · B?
-
A.
a + 2b + 3c
-
B.
a - 2b - 3c
-
C.
a * b * c
-
D.
a^2 + b^2 + c^2
Solution
A · B = a*1 + b*2 + c*3 = a + 2b + 3c.
Correct Answer: A — a + 2b + 3c
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Q. If A = (x, y) and B = (y, x), what is the scalar product A · B?
-
A.
x^2 + y^2
-
B.
xy
-
C.
x^2 - y^2
-
D.
0
Solution
A · B = x*y + y*x = 2xy.
Correct Answer: A — x^2 + y^2
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Q. If A = (x, y, z) and B = (1, 1, 1) such that A · B = 6, find x + y + z.
Solution
A · B = x + y + z = 6. Thus, x + y + z = 6.
Correct Answer: B — 6
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Q. If A = (x, y, z) and B = (1, 1, 1), find the scalar product A · B.
-
A.
x + y + z
-
B.
x - y + z
-
C.
x + y - z
-
D.
x - y - z
Solution
A · B = x*1 + y*1 + z*1 = x + y + z.
Correct Answer: A — x + y + z
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Q. If A = (x, y, z) and B = (1, 2, 3), and A · B = 14, find the value of x + y + z.
Solution
A · B = x*1 + y*2 + z*3 = 14; Let x + y + z = k; We can find values satisfying this equation.
Correct Answer: C — 7
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Q. If A = (x, y, z) and B = (2, 2, 2), and A · B = 12, what is the value of x + y + z?
Solution
A · B = 2x + 2y + 2z = 12, thus x + y + z = 6.
Correct Answer: A — 6
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Q. If A = (x, y, z) and B = (2, 3, 4), and A · B = 10, find the value of x + y + z.
Solution
x*2 + y*3 + z*4 = 10. One possible solution is x = 1, y = 1, z = 1, so x + y + z = 3.
Correct Answer: C — 3
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Q. If A = (x, y, z) and B = (2, 3, 4), find the value of x if A · B = 10.
Solution
A · B = x*2 + y*3 + z*4 = 10. If y = 0 and z = 0, then x = 5.
Correct Answer: B — 2
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Q. If A = 1i + 1j + 1k and B = 2i + 2j + 2k, what is A · B?
Solution
A · B = (1)(2) + (1)(2) + (1)(2) = 2 + 2 + 2 = 6.
Correct Answer: A — 6
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Q. If A = 2i - j and B = -i + 3j, what is the value of A · B?
Solution
A · B = (2)(-1) + (-1)(3) = -2 - 3 = -5.
Correct Answer: A — -1
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Q. If A = 6i + 8j and B = 3i + 4j, what is the scalar product A · B?
Solution
A · B = (6)(3) + (8)(4) = 18 + 32 = 50.
Correct Answer: B — 54
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Q. If A = 7i + 24j and B = 24i - 7j, calculate A · B.
Solution
A · B = (7)(24) + (24)(-7) = 168 - 168 = 0.
Correct Answer: A — 0
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Q. If A = i + 2j + 3k and B = 4i + 5j + 6k, find A · B.
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]
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B.
[1, 0], [0, 1]
-
C.
[0, 0], [0, 0]
-
D.
[6, 8], [12, 15]
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
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?
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A.
[6, 8], [10, 12]
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B.
[1, 2], [3, 4]
-
C.
[5, 6], [7, 8]
-
D.
[8, 10], [10, 12]
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]
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B.
[1, 2], [3, 4]
-
C.
[10, 13], [22, 29]
-
D.
[-1, -2], [-3, -4]
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.
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]
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]]
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?
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?
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]]
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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