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Q. Calculate the determinant of G = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]. (2022)

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

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)

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

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)

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

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, 2], [1, 0, 1]]. (2020)

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

Solution

Det(H) = 1(1*1 - 2*0) - 2(0*1 - 2*1) + 1(0*0 - 1*1) = 1(1) - 2(-2) + 1(-1) = 1 + 4 - 1 = 4.

Correct Answer: A — 0

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Q. Calculate the determinant of H = [[1, 2, 1], [0, 1, 3], [1, 0, 1]]. (2023)

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

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)

A. -5
B. 5
C. 0
D. 10

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 = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. (2021)

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

Solution

Det(H) = 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. Calculate the determinant of H = [[3, 2], [1, 4]]. (2021)

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

Solution

Determinant of H = (3*4) - (2*1) = 12 - 2 = 10.

Correct Answer: A — 10

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Q. Calculate the determinant of H = [[5, 4], [2, 3]]. (2021)

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

Solution

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

Correct Answer: B — 8

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Q. Calculate the determinant of I = [[1, 2, 1], [0, 1, 2], [1, 0, 1]]. (2021)

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

Solution

Det(I) = 1(1*1 - 2*0) - 2(0*1 - 2*1) + 1(0*0 - 1*1) = 1(1) - 2(-2) + 1(-1) = 1 + 4 - 1 = 4.

Correct Answer: B — 1

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Q. Calculate the determinant of I = [[2, 3, 1], [1, 0, 2], [0, 1, 3]]. (2015)

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

Solution

Det(I) = 2(0*3 - 2*1) - 3(1*3 - 2*0) + 1(1*1 - 0*0) = 2(0 - 2) - 3(3) + 1(1) = -4 - 9 + 1 = -12.

Correct Answer: C — 3

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Q. Calculate the determinant of J = [[1, 2, 1], [0, 1, 2], [1, 0, 1]]. (2023)

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

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 J = [[5, 6], [7, 8]]. (2014)

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

Solution

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

Correct Answer: A — -2

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Q. Calculate the determinant of the matrix F = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. (2022)

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

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 J = [[1, 2, 1], [0, 1, 0], [2, 3, 1]]. (2023)

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

Solution

The determinant of J is calculated as 1*(1*1 - 0*3) - 2*(0*1 - 0*2) + 1*(0*3 - 1*2) = 1 - 0 - 2 = -1.

Correct Answer: B — 1

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Q. Calculate the determinant of the matrix [[1, 2], [3, 4]].

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

Solution

Determinant = (1*4) - (2*3) = 4 - 6 = -2.

Correct Answer: A — -2

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Q. Calculate the determinant of the matrix \( B = \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \).

A. -1
B. 1
C. 7
D. 10

Solution

The determinant is calculated as \( 2*7 - 3*5 = 14 - 15 = -1 \).

Correct Answer: D — 10

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Q. Calculate the determinant of the matrix \( C = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \). (2020)

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

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 \( G = \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 1 \\ 3 & 1 & 2 \end{pmatrix} \).

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

Solution

The determinant is calculated as \( 2(0*2 - 1*1) - 1(1*2 - 3*1) + 3(1*1 - 3*0) = 0 \).

Correct Answer: A — -1

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Q. Calculate the determinant of the matrix \( G = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} \). (2020)

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

Solution

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

Correct Answer: A — 5

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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)

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

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. Calculate the determinant of the matrix \( I = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \). (2023)

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

Solution

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

Correct Answer: A — 10

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Q. Calculate the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \).

A. 5
B. 10
C. 7
D. 8

Solution

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

Correct Answer: A — 5

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Q. Calculate the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \).

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

Solution

The determinant is calculated as (2*7) - (3*5) = 14 - 15 = -1.

Correct Answer: A — 1

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Q. Calculate the determinant of the matrix: | 1 1 1 | | 2 2 2 | | 3 3 3 |

A. 0
B. 3
C. 6
D. 9

Solution

The rows are linearly dependent, hence the determinant is 0.

Correct Answer: A — 0

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Q. Calculate the determinant of \( D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \). (2021)

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

Solution

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

Correct Answer: A — 1

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Q. Calculate the determinant of \( D = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \). (2021)

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

Solution

The determinant of this matrix is 0 because the rows are linearly dependent.

Correct Answer: A — 0

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Q. Calculate the determinant of \( I = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \). (2022)

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

Solution

The determinant is calculated as \( 3*4 - 2*1 = 12 - 2 = 10 \).

Correct Answer: B — 8

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Q. Calculate the determinant \( \begin{vmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 4 & 1 \end{vmatrix} \).

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

Solution

The determinant is 0 because the rows are linearly dependent.

Correct Answer: A — 0

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Q. Calculate the determinant \( \begin{vmatrix} 2 & 3 \\ 5 & 7 \end{vmatrix} \)

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

Solution

The determinant is \( 2*7 - 3*5 = 14 - 15 = -1 \).

Correct Answer: A — 1

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Major Competitive Exams

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