Linear Algebra MCQs Question and Answer in Engineering Maths Page-4 section-1

31.
The sum of Eigen values of matrix, [M] is where \[\left[ {\text{M}} \right] = \left[ {\begin{array}{*{20}{c}} {215}&{650}&{795} \\ {655}&{150}&{835} \\ {485}&{355}&{550} \end{array}} \right]\]

A. 915

B. 1355

C. 1640

D. 2180

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32.
The set of equations
x + y + z = 1
ax - ay + 3z = 5
5x - 3y + az = 6
has infinite solution if a = ?

A. -4

B. -3

C. 3

D. 4

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33.
Consider the system of equations A_{(n × n)} X_{(n × 1)} = λ_{(n × 1)} where, λ is a scalar. Let (λ_i, x_i) be an eigen-pair of an eigen value and its corresponding eigen vector for real matrix A. Let $$I$$ be a (n × n) unit matrix. Which one of the following statement is NOT correct?

A. For a homogeneous n × n system of linear equations, (A - λ$$I$$)x = 0 having a nontrivial solution, the rank of (A - λ$$I$$) is less than n

B. For matrix A^m, m being a positive integer, (λ_i^m, x_i^m) will be the eigen-pair for all i

C. If A^T = A^-1, then |λ_i| = 1 for all i

D. If A^T = A, then λ_i is real for all i

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34.
The value of x for which all the eigen-values of the matrix given below are real is \[\left[ {\begin{array}{*{20}{c}} {10}&{5 + {\text{j}}}&4 \\ {\text{x}}&{20}&2 \\ 4&2&{ - 10} \end{array}} \right]\]

A. 5 + j

B. 5 - j

C. 1 - 5j

D. 1 + 5j

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35.
The system of linear equations \[\left[ {\begin{array}{{20}{c}} 2&1&3 \\ 3&0&1 \\ 1&2&5 \end{array}} \right]\left[ {\begin{array}{{20}{c}} {\text{a}} \\ {\text{b}} \\ {\text{c}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 5 \\ { - 4} \\ {14} \end{array}} \right]\] has

A. a unique solution

B. infinitely many solutions

C. no solution

D. exactly two solutions

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36.
The matrix \[{\text{P}} = \left[ {\begin{array}{{20}{c}} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}} \right]\] rotates a vector about the axis \[\left[ {\begin{array}{{20}{c}} 1 \\ 1 \\ 1 \end{array}} \right]\] by angle of

A. 30°

B. 60°

C. 90°

D. 120°

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37.
If a square matrix of order 100 has exactly 15 distinct eigen values, the degree of the minimal polynomial is

A. At least 15

B. At most 15

C. Always 15

D. Exactly 100

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38.
A matrix has eigen values -1 and -2. The corresponding eigen vectors are \[\left[ {\begin{array}{{20}{c}} 1 \\ { - 1} \end{array}} \right]\] and \[\left[ {\begin{array}{{20}{c}} 1 \\ { - 2} \end{array}} \right]\] respectively. The matrix is

A. \[\left[ {\begin{array}{*{20}{c}} 1&1 \\ { - 1}&{ - 2} \end{array}} \right]\]

B. \[\left[ {\begin{array}{*{20}{c}} 1&2 \\ { - 2}&{ - 4} \end{array}} \right]\]

C. \[\left[ {\begin{array}{*{20}{c}} { - 1}&0 \\ 0&{ - 2} \end{array}} \right]\]

D. \[\left[ {\begin{array}{*{20}{c}} 0&1 \\ { - 2}&{ - 3} \end{array}} \right]\]

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39.
Let A, B, C, D be n × n matrices, each with nonzero determinant, If ABCD = $$I$$, then B^-1 is

A. D^-1C^-1A^-1

B. CDA

C. ADC

D. does not necessarily exist

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40.
The eigen vectors of the matrix \[\left[ {\begin{array}{{20}{c}} 1&2 \\ 0&2 \end{array}} \right]\] are written in the form \[\left[ {\begin{array}{{20}{c}} 1 \\ {\text{a}} \end{array}} \right]\] and \[\left[ {\begin{array}{*{20}{c}} 1 \\ {\text{b}} \end{array}} \right].\] What is a + b = ?

A. 0

B. \[\frac{1}{2}\]

C. 1

D. 2

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Linear Algebra MCQs Question and Answer in Engineering Maths

31. The sum of Eigen values of matrix, [M] is where \[\left[ {\text{M}} \right] = \left[ {\begin{array}{*{20}{c}} {215}&{650}&{795} \\ {655}&{150}&{835} \\ {485}&{355}&{550} \end{array}} \right]\]

Answer & Solution

32. The set of equations x + y + z = 1 ax - ay + 3z = 5 5x - 3y + az = 6 has infinite solution if a = ?

Answer & Solution

33. Consider the system of equations A(n × n) X(n × 1) = λ(n × 1) where, λ is a scalar. Let (λi, xi) be an eigen-pair of an eigen value and its corresponding eigen vector for real matrix A. Let $$I$$ be a (n × n) unit matrix. Which one of the following statement is NOT correct?

Answer & Solution

34. The value of x for which all the eigen-values of the matrix given below are real is \[\left[ {\begin{array}{*{20}{c}} {10}&{5 + {\text{j}}}&4 \\ {\text{x}}&{20}&2 \\ 4&2&{ - 10} \end{array}} \right]\]

Answer & Solution

35. The system of linear equations \[\left[ {\begin{array}{*{20}{c}} 2&1&3 \\ 3&0&1 \\ 1&2&5 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\text{a}} \\ {\text{b}} \\ {\text{c}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 5 \\ { - 4} \\ {14} \end{array}} \right]\] has

Answer & Solution

36. The matrix \[{\text{P}} = \left[ {\begin{array}{*{20}{c}} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}} \right]\] rotates a vector about the axis \[\left[ {\begin{array}{*{20}{c}} 1 \\ 1 \\ 1 \end{array}} \right]\] by angle of

Answer & Solution

37. If a square matrix of order 100 has exactly 15 distinct eigen values, the degree of the minimal polynomial is

Answer & Solution

38. A matrix has eigen values -1 and -2. The corresponding eigen vectors are \[\left[ {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right]\] and \[\left[ {\begin{array}{*{20}{c}} 1 \\ { - 2} \end{array}} \right]\] respectively. The matrix is

Answer & Solution

39. Let A, B, C, D be n × n matrices, each with nonzero determinant, If ABCD = $$I$$, then B-1 is

Answer & Solution

40. The eigen vectors of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&2 \\ 0&2 \end{array}} \right]\] are written in the form \[\left[ {\begin{array}{*{20}{c}} 1 \\ {\text{a}} \end{array}} \right]\] and \[\left[ {\begin{array}{*{20}{c}} 1 \\ {\text{b}} \end{array}} \right].\] What is a + b = ?

Answer & Solution

Read More Section(Linear Algebra)

31.
The sum of Eigen values of matrix, [M] is where \[\left[ {\text{M}} \right] = \left[ {\begin{array}{*{20}{c}} {215}&{650}&{795} \\ {655}&{150}&{835} \\ {485}&{355}&{550} \end{array}} \right]\]

32.
The set of equations
x + y + z = 1
ax - ay + 3z = 5
5x - 3y + az = 6
has infinite solution if a = ?

33.
Consider the system of equations A_{(n × n)} X_{(n × 1)} = λ_{(n × 1)} where, λ is a scalar. Let (λ_i, x_i) be an eigen-pair of an eigen value and its corresponding eigen vector for real matrix A. Let $$I$$ be a (n × n) unit matrix. Which one of the following statement is NOT correct?

34.
The value of x for which all the eigen-values of the matrix given below are real is \[\left[ {\begin{array}{*{20}{c}} {10}&{5 + {\text{j}}}&4 \\ {\text{x}}&{20}&2 \\ 4&2&{ - 10} \end{array}} \right]\]

35.
The system of linear equations \[\left[ {\begin{array}{{20}{c}} 2&1&3 \\ 3&0&1 \\ 1&2&5 \end{array}} \right]\left[ {\begin{array}{{20}{c}} {\text{a}} \\ {\text{b}} \\ {\text{c}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 5 \\ { - 4} \\ {14} \end{array}} \right]\] has

36.
The matrix \[{\text{P}} = \left[ {\begin{array}{{20}{c}} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}} \right]\] rotates a vector about the axis \[\left[ {\begin{array}{{20}{c}} 1 \\ 1 \\ 1 \end{array}} \right]\] by angle of

37.
If a square matrix of order 100 has exactly 15 distinct eigen values, the degree of the minimal polynomial is

38.
A matrix has eigen values -1 and -2. The corresponding eigen vectors are \[\left[ {\begin{array}{{20}{c}} 1 \\ { - 1} \end{array}} \right]\] and \[\left[ {\begin{array}{{20}{c}} 1 \\ { - 2} \end{array}} \right]\] respectively. The matrix is

39.
Let A, B, C, D be n × n matrices, each with nonzero determinant, If ABCD = $$I$$, then B^-1 is

40.
The eigen vectors of the matrix \[\left[ {\begin{array}{{20}{c}} 1&2 \\ 0&2 \end{array}} \right]\] are written in the form \[\left[ {\begin{array}{{20}{c}} 1 \\ {\text{a}} \end{array}} \right]\] and \[\left[ {\begin{array}{*{20}{c}} 1 \\ {\text{b}} \end{array}} \right].\] What is a + b = ?