Matrices & Determinants
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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Q. If F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], what is det(F)? (2021)
Solution
Det(F) = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.
Correct Answer: A — -14
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Q. If F = [[2, 0], [0, 3]], what is det(F)? (2020)
Solution
The determinant of F is calculated as (2*3) - (0*0) = 6.
Correct Answer: B — 6
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Q. If F = [[2, 1, 3], [1, 0, 2], [0, 1, 1]], what is det(F)? (2023)
Solution
Det(F) = 2(0*1 - 2*1) - 1(1*1 - 2*0) + 3(1*1 - 0*0) = 2(0 - 2) - 1(1) + 3(1) = -4 - 1 + 3 = -2.
Correct Answer: C — 3
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Q. If F = [[2, 1, 3], [1, 0, 2], [3, 1, 1]], find det(F). (2022)
Solution
Using the determinant formula, det(F) = 2(0*1 - 2*1) - 1(1*1 - 2*3) + 3(1*1 - 0*3) = 2(0 - 2) - 1(1 - 6) + 3(1 - 0) = -4 + 5 + 3 = 4.
Correct Answer: A — -4
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Q. If F = [[2, 1, 3], [1, 0, 2], [3, 4, 1]], find det(F). (2022)
Solution
Using the determinant formula, det(F) = 2*(0*1 - 2*4) - 1*(1*1 - 2*3) + 3*(1*4 - 0*3) = 2*(-8) - 1*(-5) + 3*4 = -16 + 5 + 12 = 1.
Correct Answer: A — -10
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Q. If F = [[2, 1], [1, 3]], what is the value of det(F)? (2022)
Solution
The determinant of F is (2*3) - (1*1) = 6 - 1 = 5.
Correct Answer: A — 5
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Q. If H = [[1, 2, 1], [0, 1, 0], [2, 1, 1]], find det(H). (2021)
Solution
The determinant of H is calculated as 1(1*1 - 0*1) - 2(0*1 - 0*2) + 1(0*1 - 1*2) = 1 - 0 - 2 = -1.
Correct Answer: B — 1
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Q. If H = [[1, 2], [2, 4]], what is det(H)? (2020)
Solution
The determinant of H is (1*4) - (2*2) = 4 - 4 = 0.
Correct Answer: A — 0
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Q. If H = [[2, 3], [4, 5]], find det(H). (2022)
Solution
Det(H) = (2*5) - (3*4) = 10 - 12 = -2.
Correct Answer: D — 7
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Q. If I = [[1, 0, 2], [0, 1, 3], [1, 0, 4]], find det(I). (2021)
Solution
Using cofactor expansion, det(I) = 1(1*4 - 3*0) - 0 + 2(0*0 - 1*1) = 4 - 2 = 2.
Correct Answer: B — 2
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Q. If I = [[1, 0, 2], [0, 1, 3], [1, 1, 0]], find det(I). (2023)
Solution
Using the determinant formula for 3x3 matrices, det(I) = 1(1*0 - 3*1) - 0(0 - 3*1) + 2(0 - 1*1) = 0 - 0 - 2 = -2.
Correct Answer: A — -1
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Q. If I = [[1, 2], [2, 4]], what is det(I)? (2021)
Solution
The determinant of I is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. If J = [[1, 1], [1, 1]], what is det(J)? (2019)
Solution
Det(J) = (1*1) - (1*1) = 1 - 1 = 0.
Correct Answer: A — 0
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Q. If J = [[1, 2, 1], [0, 1, 3], [2, 1, 0]], calculate det(J). (2023)
Solution
Using the determinant formula, det(J) = 1*(1*0 - 3*1) - 2*(0*0 - 3*2) + 1*(0*1 - 1*2) = 1*(-3) - 2*(-6) + 1*(-2) = -3 + 12 - 2 = 7.
Correct Answer: A — -4
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Q. If J = [[1, 2], [2, 4]], what is det(J)? (2022)
Solution
Det(J) = (1*4) - (2*2) = 4 - 4 = 0.
Correct Answer: A — 0
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Q. If \( B = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \), what is \( |B| \)? (2022)
Solution
The determinant of \( B \) is \( 2*4 - 3*1 = 8 - 3 = 5 \).
Correct Answer: A — 5
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Q. What is the determinant of G = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]? (2023)
Solution
Det(G) = 1(2*6 - 3*3) - 1(1*6 - 1*3) + 1(1*3 - 1*2) = 1(12 - 9) - 1(6 - 3) + 1(3 - 2) = 3 - 3 + 1 = 1.
Correct Answer: A — 0
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Q. What is the determinant of G = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]? (2020)
Solution
Det(G) = 1(5*9 - 6*8) - 2(4*9 - 6*7) + 3(4*8 - 5*7) = 1(45 - 48) - 2(36 - 42) + 3(32 - 35) = -3 + 12 - 9 = 0.
Correct Answer: A — 0
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Q. What is 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. What is the determinant of G = [[2, 1], [1, 2]]? (2021)
Solution
Det(G) = (2*2) - (1*1) = 4 - 1 = 3.
Correct Answer: B — 2
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Q. What is the determinant of G = [[2, 2], [2, 2]]? (2020)
Solution
Det(G) = (2*2) - (2*2) = 4 - 4 = 0.
Correct Answer: A — 0
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Q. What is the determinant of G = [[3, 2], [1, 4]]? (2022)
Solution
The determinant of G is (3*4) - (2*1) = 12 - 2 = 10.
Correct Answer: A — 10
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Q. What is the determinant of H = [[3, 1], [2, 4]]? (2023)
Solution
The determinant of H is calculated as (3*4) - (1*2) = 12 - 2 = 10.
Correct Answer: A — 10
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Q. What is the determinant of H = [[3, 2], [1, 4]]? (2020)
Solution
The determinant of H is (3*4) - (2*1) = 12 - 2 = 10.
Correct Answer: A — 10
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Q. What is the determinant of H = [[3, 2], [1, 5]]? (2021)
Solution
Determinant of H = (3*5) - (2*1) = 15 - 2 = 13.
Correct Answer: A — 7
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Q. What is the determinant of J = [[2, 3], [4, 5]]? (2020)
Solution
The determinant of J is (2*5) - (3*4) = 10 - 12 = -2.
Correct Answer: D — 10
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Q. What is the determinant of J = [[5, 4], [2, 3]]? (2022)
Solution
The determinant of J is (5*3) - (4*2) = 15 - 8 = 7.
Correct Answer: A — 7
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Q. What is the determinant of the identity matrix I = [[1, 0], [0, 1]]? (2021)
Solution
Det(I) = (1*1) - (0*0) = 1 - 0 = 1.
Correct Answer: B — 1
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Q. What is the determinant of the matrix E = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]? (2021)
Solution
The determinant of E is calculated using the rule of Sarrus or cofactor expansion, resulting in -14.
Correct Answer: A — -14
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