Linear Algebra MCQs Question and Answer in Engineering Maths

71

Consider a non-homogeneous system of linear equations representing mathematically an overdetermined system. Such a system will be

A) consistent having a unique solution B) consistent having many solutions C) inconsistent having a unique solution D) inconsistent having no solution

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Answer: Option B

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72

The dimension of the null space of the matrix \[\left[ {\begin{array}{*{20}{c}} 0&1&1 \\ 1&{ - 1}&0 \\ { - 1}&0&1 \end{array}} \right]\] is

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

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Answer: Option B

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73

X and Y are non-zero square matrices of size n × n. If XY = 0_{n × n} then

A) |X| = 0 and |Y| ≠ 0 B) |X| ≠ 0 and |Y| = 0 C) |X| = 0 and |Y| = 0 D) |X| ≠ 0 and |Y| ≠ 0

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Answer: Option C

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74

A is m × n full rank matrix with m > n and $$I$$ is an identity matrix. Let matrix A' = (A^TA)^-1A^T, Then, which one of the following statement is TRUE?

A) AA' A = A B) (AA')² = A C) AA'A = $$I$$ D) AA'A = A'

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Answer: Option A

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75

The minimum and the maximum eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&1&3 \\ 1&5&1 \\ 3&1&1 \end{array}} \right]\] are -2 and 6, respectively. What is the other eigen value?

A) 5 B) 3 C) 1 D) -1

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Answer: Option B

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76

Consider the matrix \[\left[ {\begin{array}{*{20}{c}} 5&{ - 1} \\ 4&1 \end{array}} \right]\] . Which one of the following statements is TRUE for the eigen values and eigen vectors of this matrix?

A) Eigen value 3 has a multiplicity of 2, and only one independent eigen vector exists B) Eigen value 3 has a multiplicity of 2, and two independent eigen vector exists C) Eigen value 3 has a multiplicity of 2, and no independent eigen vector exists D) Eigen value are 3 and -3, and two independent eigen vectors exist

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Answer: Option A

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77

Let M be a real 4 × 4 matrix. Consider the following statements:
S1: M has 4 linearly independent eigenvectors.
S2: M has 4 distinct eigenvalues.
S3: M is non-singular (invertible).
Which one among the following is TRUE?

A) S1 implies S2 B) S1 implies S3 C) S2 implies S1 D) S3 implies S2

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Answer: Option C

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78

The smallest and largest Eigen values of the following matrix are \[\left[ {\begin{array}{*{20}{c}} 3&{ - 2}&2 \\ 4&{ - 4}&6 \\ 2&{ - 3}&5 \end{array}} \right]\]

A) 1.5 and 2.5 B) 0.5 and 2.5 C) 1.0 and 3.0 D) 1.0 and 2.0

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Answer: Option D

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79

Eigen values of a matrix \[{\text{S}} = \left[ {\begin{array}{*{20}{c}} 3&2 \\ 2&3 \end{array}} \right]\] are 5 and 1. What are the eigen values of the matrix S² = SS?

A) 1 and 25 B) 6 and 4 C) 5 and 1 D) 2 and 10

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Answer: Option A

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80

If any two columns of a determinant \[{\text{P}} = \left| {\begin{array}{*{20}{c}} 4&7&8 \\ 3&1&5 \\ 9&6&2 \end{array}} \right|\] are interchanged, which one of the following statements regarding the value of the determinant is CORRECT?

A) Absolute value remains unchanged but sign will change B) Both absolute value and sign will change C) Absolute value will change but sign will not change D) Both absolute value and sign will remain unchanged

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Answer: Option A

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