Linear Algebra MCQs Question and Answer in Engineering Maths

1

Let A be an n × n real matrix such that A² = $$I$$ and y be an n-dimensional vector.
Then the linear system of equations Ax = y has

A) no solution B) a unique solution C) more than one but finitely many independent solutions D) infinitely many independent solutions

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Answer & Solution

Answer: Option B

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2

Let A = [a_ij],1 ≤ i, j ≤ n with n ≥ 3 and a_ij = i.j. The rank of A is

A) 0 B) 1 C) n - 1 D) n

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Answer & Solution

Answer: Option B

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3

All the four entries of the 2 × 2 matrix \[{\text{P}} = \left[ {\begin{array}{*{20}{c}} {{{\text{p}}_{11}}}&{{{\text{p}}_{12}}} \\ {{{\text{p}}_{21}}}&{{{\text{p}}_{22}}} \end{array}} \right]\] are nonzero, and one of its eigen values is zero. Which of the following statements is true?

A) p₁₁p₂₂ - p₁₂p₂₁ = 1 B) p₁₁p₂₂ - p₁₂p₂₁ = -1 C) p₁₁p₂₂ - p₁₂p₂₁ = 0 D) p₁₁p₂₂ + p₁₂p₂₁ = 0

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Answer & Solution

Answer: Option C

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4

The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 0&1 \\ { - 1}&0 \end{array}} \right]\] are

A) i and -i B) 1 and -1 C) 0 and 1 D) 0 and -1

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Answer & Solution

Answer: Option A

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5

If the rank of a (5 × 6) matrix Q is 4, then which one of the following statements is correct?

A) Q will have four linearly independent rows and four linearly independent columns B) Q will have four linearly independent rows and five linearly independent columns C) QQ^T will be invertible D) Q^TQ will be invertible

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Answer & Solution

Answer: Option A

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6

Consider the following system of equations
2x₁ + x₂ + x₃ = 0
x₂ - x₃ = 0
x₁ + x₂ = 0
This system has

A) a unique solution B) no solution C) infinite number of solutions D) five solutions

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Answer & Solution

Answer: Option C

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7

Let A be an n × n matrix with rank r(0 < r < n). Then AX = 0 has p independent solutions, where p is

A) r B) n C) n - r D) n + r

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Answer & Solution

Answer: Option C

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8

The maximum value of "a" such that the matrix \[\left( {\begin{array}{*{20}{c}} { - 3}&0&{ - 2} \\ 1&{ - 1}&0 \\ 0&{\text{a}}&{ - 2} \end{array}} \right)\] has three linearly independent real eigen vectors is

A) \[\frac{2}{{3\sqrt 3 }}\] B) \[\frac{1}{{3\sqrt 3 }}\] C) \[\frac{{1 + 2\sqrt 3 }}{{3\sqrt 3 }}\] D) \[\frac{{1 + \sqrt 3 }}{{3\sqrt 3 }}\]

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Answer & Solution

Answer: Option B

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9

In the matrix equation Px = q, which of the following is a necessary condition for the existence of at least one solution for the unknown vector x

A) Augmented matrix [Pq] must have the same rank as matrix P B) Vector q must have only non-zero elements C) Matrix P must be singular D) Matrix P must be square

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Answer & Solution

Answer: Option A

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10

If the system
2x - y + 3z = 2
x + y + 2z = 2
5x - y + az = b
has infinitely many solutions, then the values of a and b, respectively, are

A) -8 and 6 B) 8 and 6 C) -8 and -6 D) 8 and -6

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Answer & Solution

Answer: Option B

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