Linear Algebra MCQs with Answers 2027

Q61

Fora given matrix

{\text{A}} = \left[ {\begin{array}{*{20}{c}} 2&{ - 2}&3 \\ { - 2}&{ - 1}&6 \\ 1&2&0 \end{array}} \right],

one of the eigen values is 3. The other two eigen values are

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Q62

Open question

Let A be n × n real valued square symmetric matrix of rank 2 with

i = 1 \sum n j = 1 \sum n A_{ij}^{2} = 50.

Consider the following statements.
I. One eigen value must be in [-5, 5]
II. The eigen value with the largest magnitude must be strictly greater than 5.
Which of the above statements about eigen values of A is/are necessarily CORRECT?

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Q63

Open question

Real matrices [A]_3×1, [B]_3×3, [C]_3×5, [D]_5×3, [E]_5×5 and [F]_5×1 are given. Matrices [B] and [E] are symmetric.
Following statements are made with respect to these matrices.
1. Matrix product [F]^T[C]^T[B] [C] [F] is a scalar.
2. Matrix product [D]^T[F] [D] is always symmetric.
With reference to above statements, which of the following applies?

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Q64

Open question

Let the Eigen vector of the matrix

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

be 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 the value of (a + b) = ?

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Q65

Open question

Which one of the following matrices is singular?

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Q66

Open question

Let c₁ ..... c_n be scalars, not all zero, such that

i = 1 \sum n c_{i} a_{i} = 0

where a_i are column vectors in Rⁿ. Consider the set of linear equations Ax = b where A = [a₁ ..... a_n] and

b = i = 1 \sum n a_{i} .

The set of equations has

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Q67

Open question

Two matrices A and B are given below:

{\text{A}} = \left[ {\begin{array}{*{20}{c}} {\text{p}}&{\text{q}} \\ {\text{r}}&{\text{s}} \end{array}} \right]{\text{;}}\,{\text{B}} = \left[ {\begin{array}{*{20}{c}} {{{\text{p}}^2} + {{\text{q}}^2}}&{{\text{pr}} + {\text{qs}}} \\ {{\text{pr}} + {\text{qs}}}&{{{\text{r}}^2} + {{\text{s}}^2}} \end{array}} \right]

If the rank of matrix A is N, then the rank of matrix B is

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Q68

Open question

Consider the matrix

{\text{P}} = \left[ {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}}&0&{\frac{1}{{\sqrt 2 }}} \\ 0&1&0 \\ {\frac{{ - 1}}{{\sqrt 2 }}}&0&{\frac{1}{{\sqrt 2 }}} \end{array}} \right]

Which one of the following statements about P is INCORRECT?

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Q69

Open question

The eigen values and the corresponding eigen vectors of a 2 × 2 matrix are given by

\begin{array}{*{20}{c}} {{\text{Eigen value}}}&{{\text{Eigen vector}}} \\ {{\lambda _1} = 8}&{{{\text{v}}_1} = \left[ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right]} \\ {{\lambda _2} = 4}&{{{\text{v}}_2} = \left[ {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right]} \end{array}

The matrix is

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Q70

Open question

A scalar valued function is defined as f(X) = X^TAX + b^TX + c, where A is a symmetric positive definite matrix with dimension n × n; b and x are vectors of dimension n × 1. The minimum value of f(X) will occur when X equals

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Q71

Open question

The minimum eigen value of the following matrix is

\left[ {\begin{array}{*{20}{c}} 3&5&2 \\ 5&{12}&7 \\ 2&7&5 \end{array}} \right]

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Q72

Open question

With reference to the conventional Cartesian (x, y) coordinate system, the vertices of a triangle have the following coordinates; (x₁, y₁) = (1, 0); (x₂, y₂) = (2, 2); (x₃, y₃) = (4, 3). The area of the triangle is equal to