Q. If A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]], what is A + B?
A.
[[1, 3], [4, 4]]
B.
[[1, 2], [3, 5]]
C.
[[1, 3], [2, 4]]
D.
[[1, 2], [4, 4]]
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Solution
The sum A + B is calculated as [[1+0, 2+1], [3+1, 4+0]] = [[1, 3], [4, 4]].
Correct Answer:
A
— [[1, 3], [4, 4]]
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Q. If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], what is the product AB?
A.
[19, 22], [43, 50]
B.
[23, 28], [31, 36]
C.
[17, 20], [39, 46]
D.
[29, 34], [55, 64]
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Solution
The product AB is calculated as [[(1*5 + 2*7), (1*6 + 2*8)], [(3*5 + 4*7), (3*6 + 4*8)]] = [[19, 22], [43, 50]].
Correct Answer:
A
— [19, 22], [43, 50]
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Q. If the matrix A = [[1, 0], [0, 1]] is the identity matrix, what is A^2?
A.
[[1, 0], [0, 1]]
B.
[[0, 1], [1, 0]]
C.
[[1, 1], [1, 1]]
D.
[[0, 0], [0, 0]]
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Solution
The identity matrix squared is still the identity matrix: A^2 = A.
Correct Answer:
A
— [[1, 0], [0, 1]]
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Q. If the matrix A = [[2, 3], [5, 7]], what is the eigenvalue of A?
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Solution
The eigenvalues are found by solving the characteristic polynomial det(A - λI) = 0. The eigenvalues are 9 and 1.
Correct Answer:
B
— 9
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Q. What is the determinant of the matrix [[2, 3], [1, 4]]?
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Solution
The determinant is calculated as (2*4) - (3*1) = 8 - 3 = 5.
Correct Answer:
A
— 5
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Q. What is the inverse of the matrix [[4, 7], [2, 6]]?
A.
[[3, -3.5], [-1, 2]]
B.
[[6, -7], [-2, 4]]
C.
[[6, -7], [-2, 4]]
D.
[[1.5, -1.75], [-0.5, 1]]
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Solution
The inverse is calculated as (1/det(A)) * adj(A). The determinant is 10, and the adjugate is [[6, -7], [-2, 4]]. Thus, the inverse is [[6/10, -7/10], [-2/10, 4/10]] = [[3/5, -7/10], [-1/5, 2/5]].
Correct Answer:
A
— [[3, -3.5], [-1, 2]]
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Q. What is the rank of the matrix [[1, 2, 3], [0, 0, 0], [4, 5, 6]]?
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Solution
The rank is the maximum number of linearly independent row vectors. Here, there are 2 non-zero rows, so the rank is 2.
Correct Answer:
B
— 2
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Q. What is the result of the matrix multiplication [[1, 2], [3, 4]] * [[5], [6]]?
A.
[17]
B.
[19]
C.
[23]
D.
[29]
Show solution
Solution
The result is calculated as [[(1*5 + 2*6)], [(3*5 + 4*6)]] = [[17], [39]].
Correct Answer:
A
— [17]
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Q. What is the trace of the matrix [[3, 2, 1], [0, 5, 0], [4, 0, 6]]?
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Solution
The trace is the sum of the diagonal elements: 3 + 5 + 6 = 14.
Correct Answer:
A
— 14
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