Matrices & Determinants
Q. Calculate the determinant of D = [[2, 3, 1], [1, 0, 2], [4, 1, 0]]. (2020)
Solution
Det(D) = 2(0*0 - 2*1) - 3(1*0 - 2*4) + 1(1*1 - 0*4) = 2(0 - 2) - 3(0 - 8) + 1(1) = -4 + 24 + 1 = 21.
Correct Answer: A — -10
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Q. Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [0, 1, 3]]. (2023)
Solution
Det(D) = 3(0*3 - 2*1) - 2(1*3 - 0*2) + 1(1*1 - 0*0) = 3(0 - 2) - 2(3) + 1(1) = -6 - 6 + 1 = -11.
Correct Answer: A — 1
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Q. Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [2, 1, 3]]. (2020)
Solution
Det(D) = 3(0*3 - 2*1) - 2(1*3 - 2*2) + 1(1*1 - 0*2) = 3(0 - 2) - 2(3 - 4) + 1(1) = -6 + 2 + 1 = -3.
Correct Answer: A — 1
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Q. Calculate the determinant of D = [[4, 2], [1, 3]]. (2020)
Solution
Determinant of D = (4*3) - (2*1) = 12 - 2 = 10.
Correct Answer: A — 10
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Q. Calculate the determinant of D = [[4, 2], [3, 1]]. (2020)
Solution
Determinant of D = (4*1) - (2*3) = 4 - 6 = -2.
Correct Answer: A — -2
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Q. Calculate the determinant of F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. (2023)
Solution
Using the determinant formula, det(F) = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = -24 + 40 - 15 = 1.
Correct Answer: A — -14
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Q. Calculate the determinant of G = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]. (2022)
Solution
The determinant of G is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. Calculate the determinant of G = [[1, 2, 1], [0, 1, 0], [2, 3, 1]]. (2023)
Solution
Using cofactor expansion, det(G) = 1(1*1 - 0*3) - 2(0*1 - 0*2) + 1(0*3 - 1*2) = 1 - 0 - 2 = -1.
Correct Answer: A — -1
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Q. Calculate the determinant of G = [[1, 2, 1], [0, 1, 4], [1, 0, 0]]. (2021)
Solution
Using the determinant formula for 3x3 matrices, det(G) = 1(1*0 - 4*0) - 2(0*0 - 4*1) + 1(0*0 - 1*1) = 0 + 8 - 1 = 7.
Correct Answer: A — -2
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Q. Calculate the determinant of H = [[1, 2, 1], [0, 1, 3], [1, 0, 1]]. (2023)
Solution
Det(H) = 1(1*1 - 3*0) - 2(0*1 - 3*1) + 1(0*0 - 1*1) = 1(1) - 2(-3) + 1(-1) = 1 + 6 - 1 = 6.
Correct Answer: A — -1
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Q. Calculate the determinant of H = [[1, 2, 1], [0, 1, 3], [2, 1, 0]]. (2020)
Solution
Det(H) = 1(1*0 - 3*1) - 2(0*0 - 3*2) + 1(0*1 - 1*2) = 1(0 - 3) - 2(0 - 6) + 1(0 - 2) = -3 + 12 - 2 = 7.
Correct Answer: A — -5
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Q. Calculate the determinant of H = [[5, 4], [2, 3]]. (2021)
Solution
Det(H) = (5*3) - (4*2) = 15 - 8 = 7.
Correct Answer: B — 8
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Q. Calculate the determinant of J = [[1, 2, 1], [0, 1, 2], [1, 0, 1]]. (2023)
Solution
Det(J) = 1(1*1 - 2*0) - 2(0*1 - 1*1) + 1(0*0 - 1*1) = 1(1) - 2(-1) + 1(-1) = 1 + 2 - 1 = 2.
Correct Answer: C — 2
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Q. Calculate the determinant of the matrix F = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. (2022)
Solution
The determinant of F is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. Calculate the determinant of the matrix \( C = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \). (2020)
Solution
The determinant is \( 5*8 - 6*7 = 40 - 42 = -2 \).
Correct Answer: A — -2
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Q. Calculate the determinant of the matrix \( H = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix} \). (2020)
Solution
The determinant of an upper triangular matrix is the product of its diagonal elements: \( 1*1*1 = 1 \).
Correct Answer: A — 1
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Q. Find the determinant of E = [[3, 2], [1, 4]]. (2022)
Solution
Det(E) = (3*4) - (2*1) = 12 - 2 = 10.
Correct Answer: A — 10
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Q. Find the determinant of E = [[4, 2], [1, 3]]. (2023)
Solution
Det(E) = (4*3) - (2*1) = 12 - 2 = 10.
Correct Answer: A — 10
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Q. Find the determinant of F = [[4, 5], [6, 7]]. (2020)
Solution
Det(F) = (4*7) - (5*6) = 28 - 30 = -2.
Correct Answer: A — -2
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Q. Find the determinant of G = [[1, 2], [2, 4]]. (2020)
Solution
Determinant of G = (1*4) - (2*2) = 4 - 4 = 0.
Correct Answer: A — 0
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Q. Find the determinant of H = [[3, 1], [2, 5]]. (2021)
Solution
Determinant of H = (3*5) - (1*2) = 15 - 2 = 13.
Correct Answer: A — 7
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Q. Find the determinant of J = [[5, 2], [1, 3]]. (2020)
Solution
The determinant of J is calculated as (5*3) - (2*1) = 15 - 2 = 13.
Correct Answer: A — 10
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Q. Find the determinant of the matrix D = [[3, 2, 1], [1, 0, 2], [2, 1, 3]]. (2020)
Solution
The determinant of D can be calculated using the rule of Sarrus or cofactor expansion, which results in 0.
Correct Answer: A — 0
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Q. Find the determinant of the matrix D = [[4, 2], [3, 1]]. (2023)
Solution
The determinant of D is calculated as (4*1) - (2*3) = 4 - 6 = -2.
Correct Answer: A — -2
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Q. Find the determinant of the matrix \( E = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \). (2021)
Solution
The determinant is \( 3*4 - 2*1 = 12 - 2 = 10 \).
Correct Answer: A — 10
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Q. For the matrix E = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find det(E). (2021)
Solution
Using the determinant formula, det(E) = 1*(1*0 - 4*6) - 2*(0*0 - 4*5) + 3*(0*6 - 1*5) = 1*(-24) - 2*(-20) - 15 = -24 + 40 - 15 = 1.
Correct Answer: A — -24
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Q. For the matrix E = [[1, 2], [2, 4]], what is the determinant? (2021)
Solution
Determinant of E = (1*4) - (2*2) = 4 - 4 = 0.
Correct Answer: A — 0
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Q. If C = [[0, 1], [1, 0]], what is det(C)? (2022)
Solution
Determinant of C = (0*0) - (1*1) = 0 - 1 = -1.
Correct Answer: C — -1
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Q. If E = [[2, 1, 3], [1, 0, 2], [4, 1, 1]], what is det(E)? (2020)
Solution
The determinant of E can be calculated using the rule of Sarrus or cofactor expansion, resulting in 0.
Correct Answer: A — -1
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Q. If E = [[a, b], [c, d]], what is the expression for det(E)? (2023)
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A.
ad - bc
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B.
ab + cd
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C.
ac - bd
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D.
bc - ad
Solution
The determinant of E is calculated as (a*d) - (b*c) = ad - bc.
Correct Answer: A — ad - bc
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