Linear Algebra MCQs Question and Answer in Engineering Maths

21

The rank of the matrix \[{\text{M}} = \left[ {\begin{array}{*{20}{c}} 5&{10}&{10} \\ 1&0&2 \\ 3&6&6 \end{array}} \right]\] is

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

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Answer: Option C

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22

One of the eigen vectors of the matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 2&2 \\ 1&3 \end{array}} \right]\] is

A) \[\left\{ {\begin{array}{*{20}{c}} 2 \\ { - 1} \end{array}} \right\}\] B) \[\left\{ {\begin{array}{*{20}{c}} 2 \\ 1 \end{array}} \right\}\] C) \[\left\{ {\begin{array}{*{20}{c}} 4 \\ 1 \end{array}} \right\}\] D) \[\left\{ {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right\}\]

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Answer: Option A

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23

The lowest Eigen value of the 2 × 2 matrix \[\left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&3 \end{array}} \right]\]

A) 1 B) 2 C) 3 D) 5

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Answer: Option B

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24

Which one of the following statements is TRUE about every n × n matrix with only real eigen values?

A) If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B) If the trace of the matrix is positive, all its eigen values are positive C) If the determinant of the matrix is positive, all its eigen values are positive D) If the product of the trace and determinant of the matrix is positive, all its eigen values are positive

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Answer: Option A

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25

The equation \[\left[ {\begin{array}{*{20}{c}} 2&{ - 2} \\ 1&{ - 1} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\text{x}}_1}} \\ {{{\text{x}}_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right]\] has

A) no solution B) only one solution \[\left[ {\begin{array}{*{20}{c}} {{{\text{x}}_1}} \\ {{{\text{x}}_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right]\] C) non-zero unique solution D) multiple solutions

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Answer: Option D

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26

Consider the system of simultaneous equations
x + 2y + z = 6
2x + y + 2z = 6
x + y + z = 5
This system has

A) unique solution B) infinite number of solutions C) no solution D) exactly two solutions

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Answer: Option C

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27

The system of equations
x + y + z = 6
x + 4y + 6z = 20
x + 4y + \[\lambda \]z = \[\mu \]
has NO solution for values of \[\lambda \] and \[\mu \] given by

A) \[\lambda = 6,\,\mu = 20\] B) \[\lambda = 6,\,\mu \ne 20\] C) \[\lambda \ne 6,\,\mu = 20\] D) \[\lambda \ne 6,\,\mu \ne 20\]

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Answer: Option B

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28

Let the eigen values of a 2 × 2 matrix A be 1, -2 with eigen vectors x₁ and x₂ respectively. Then the eigen values and eigen vectors of the matrix A² - 3A + 4$$I$$ would, respectively, be

A) 2, 14; x₁, x₂ B) 2, 14; x₁ + x₂, x₁ - x₂ C) 2, 0; x₁, x₂ D) 2, 0; x₁ + x₂, x₁ - x₂

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Answer: Option A

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29

The figure shows a shape ABC and its mirror image A₁B₁C₁ across the horizontal axis (X-axis). The coordinate transformation matrix that maps ABC to A₁B₁C₁ is
Linear Algebra mcq question image

A) \[\left[ {\begin{array}{*{20}{c}} 0&1 \\ 1&0 \end{array}} \right]\] B) \[\left[ {\begin{array}{*{20}{c}} 0&1 \\ { - 1}&0 \end{array}} \right]\] C) \[\left[ {\begin{array}{*{20}{c}} { - 1}&0 \\ 0&1 \end{array}} \right]\] D) \[\left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&{ - 1} \end{array}} \right]\]

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Answer: Option D

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30

If the entries in each column of a square matrix M add up to 1, then an eiqen value of M is

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

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Answer: Option D

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