Matrices & Determinants for Competitive Exam Success

Matrices & Determinants

Q. If I = [[1, 0, 2], [0, 1, 3], [1, 0, 4]], find det(I). (2021)

A. 1
B. 2
C. 3
D. 4

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)

A. -1
B. 0
C. 1
D. 2

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)

A. 0
B. 1
C. 2
D. 3

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)

A. 0
B. 1
C. 2
D. 3

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, 0], [2, 1, 1]], find det(J). (2019)

A. 0
B. 1
C. 2
D. 3

Solution

Determinant of J = 1(1*1 - 0*1) - 2(0*1 - 0*2) + 1(0*1 - 1*2) = 1 - 0 - 2 = -1.

Correct Answer: C — 2

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Q. If J = [[1, 2, 1], [0, 1, 3], [2, 1, 0]], calculate det(J). (2023)

A. -4
B. 4
C. 0
D. 2

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)

A. 0
B. 1
C. 2
D. 4

Solution

Det(J) = (1*4) - (2*2) = 4 - 4 = 0.

Correct Answer: A — 0

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Q. If J = [[3, 1], [2, 4]], find det(J). (2023)

A. 10
B. 12
C. 8
D. 6

Solution

Det(J) = (3*4) - (1*2) = 12 - 2 = 10.

Correct Answer: A — 10

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Q. If \( B = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \), what is \( |B| \)? (2022)

A. 5
B. 10
C. 1
D. 8

Solution

The determinant of \( B \) is \( 2*4 - 3*1 = 8 - 3 = 5 \).

Correct Answer: A — 5

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Q. If \( E = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \), what is \( |E| \)? (2023)

A. 0
B. 1
C. -1
D. 2

Solution

The determinant is calculated as \( 0*0 - 1*1 = 0 - 1 = -1 \).

Correct Answer: C — -1

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Q. If \( E = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \), what is \( |E| \)? (2022)

A. 0
B. 1
C. 2
D. 3

Solution

The determinant is \( 1*1 - 1*1 = 1 - 1 = 0 \).

Correct Answer: A — 0

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Q. If \( I = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \), what is \( |I| \)? (2021)

A. 0
B. 1
C. 2
D. 3

Solution

The determinant is \( 1*1 - 1*1 = 1 - 1 = 0 \).

Correct Answer: A — 0

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Q. What is the determinant of F = [[2, 0], [0, 3]]? (2023)

A. 0
B. 2
C. 3
D. 6

Solution

Determinant of F = (2*3) - (0*0) = 6.

Correct Answer: D — 6

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Q. What is the determinant of G = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]? (2023)

A. 0
B. 1
C. 2
D. 3

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)

A. 0
B. 1
C. 2
D. 3

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)

A. 0
B. 1
C. 2
D. 4

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)

A. 1
B. 2
C. 3
D. 4

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)

A. 0
B. 4
C. 2
D. 1

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)

A. 10
B. 8
C. 6
D. 12

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 G = [[5, 4], [2, 3]]? (2020)

A. 7
B. 10
C. 11
D. 12

Solution

Det(G) = (5*3) - (4*2) = 15 - 8 = 7.

Correct Answer: B — 10

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Q. What is the determinant of G = [[5, 6], [7, 8]]? (2017)

A. -2
B. 2
C. 6
D. 8

Solution

Det(G) = (5*8) - (6*7) = 40 - 42 = -2.

Correct Answer: A — -2

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Q. What is the determinant of H = [[3, 1], [2, 4]]? (2023)

A. 10
B. 11
C. 12
D. 13

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)

A. 10
B. 11
C. 12
D. 13

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)

A. 7
B. 8
C. 10
D. 15

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 I = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]? (2015)

A. 0
B. 1
C. 2
D. 3

Solution

Det(I) = 1.

Correct Answer: B — 1

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Q. What is the determinant of I = [[5, 6], [7, 8]]? (2020)

A. -2
B. 2
C. 10
D. 12

Solution

Determinant of I = (5*8) - (6*7) = 40 - 42 = -2.

Correct Answer: A — -2

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Q. What is the determinant of J = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]? (2014)

A. -14
B. 14
C. 0
D. 10

Solution

Det(J) = 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. What is the determinant of J = [[1, 2], [3, 5]]? (2020)

A. -1
B. 1
C. 2
D. 3

Solution

Det(J) = (1*5) - (2*3) = 5 - 6 = -1.

Correct Answer: A — -1

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Q. What is the determinant of J = [[2, 3], [4, 5]]? (2020)

A. -2
B. 2
C. 7
D. 10

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)

A. 7
B. 8
C. 9
D. 10

Solution

The determinant of J is (5*3) - (4*2) = 15 - 8 = 7.

Correct Answer: A — 7

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Matrices & Determinants

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