Linear Algebra — Part 2 MCQs with Explanations

PQ = $$I$$ and QP = $$I$$ is the correct answer to “The matrix P is the inverse of a matrix Q. If $$I$$ denotes the identity matrix, which one of the following options is correct”. This question is standard in competitive examination syllabi for Linear Algebra.

BT = -B is the correct answer to “A square matrix B is skew-symmetric if”. This question is standard in competitive examination syllabi for Linear Algebra.

19A + 30$$I$$ is the correct answer to “Given that, \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} { - 5}&{ - 3} \\ 2&0 \end{array}} \right]\] and \[I = \left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right],\] the value A 3 is”. This question is standard in competitive examination syllabi for Linear Algebra.

(2 × 2) is the correct answer to “Consider the matrices X (4 × 3) , Y (4 × 3) and P (2 × 3) . The order of [P(X T Y) -1 P T ] T will be”. This question is standard in competitive examination syllabi for Linear Algebra.

2 is the correct answer to “The rank of the matrix \[\left[ {\begin{array}{*{20}{c}} { - 4}&1&{ - 1} \\ { - 1}&{ - 1}&{ - 1} \\ 7&{ - 3}&1 \end{array}} \right]\] is”. This question is standard in competitive examination syllabi for Linear Algebra.

3a + b - c = 0 is the correct answer to “Consider the following linear system. x + 2y - 3z = a 2x + 3y + 3z = b 5x + 9y - 6z = c This system is consistent if a, b and c satisfy the equation”. This question is standard in competitive examination syllabi for Linear Algebra.

0 is the correct answer to “Consider a 3 × 3 real symmetric matrix S such that two of its eigen values are a ≠ 0, b ≠ 0 with respective eigen vectors \[\left[ {\begin{array}{*{20}{c}} {{{\text{x}}_1}} \\ {{{\text{x}}_2}} \\ {{{\text{x}}_3}} \end{array}} \right],\left[ {\begin{array}{*{20}{c}} {{{\text{y}}_1}} \\ {{{\text{y}}_2}} \\ {{{\text{y}}_3}} \end{array}} \right].\] If a ≠ b then x 1 y 1 + x 2 y 2 + x 3 y 3 equals”. This question is standard in competitive examination syllabi for Linear Algebra.

4 is the correct answer to “For which value of x will the matrix given below become singular? \[\left[ {\begin{array}{*{20}{c}} 8&{\text{x}}&0 \\ 4&0&2 \\ {12}&6&0 \end{array}} \right]\]”. This question is standard in competitive examination syllabi for Linear Algebra.

zero is the correct answer to “The number of solutions of the simultaneous algebraic equation y = 3x + 3 and y = 3x + 5 is:”. This question is standard in competitive examination syllabi for Linear Algebra.

2 is the correct answer to “Given Matrix \[\left[ {\text{A}} \right] = \left[ {\begin{array}{*{20}{c}} 4&2&1&3 \\ 6&3&4&7 \\ 2&1&0&1 \end{array}} \right],\] the rank of the matrix is”. This question is standard in competitive examination syllabi for Linear Algebra.

Hermitian matrix is the correct answer to “If \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} {2 + {\text{i}}}&3&{ - 1 + 3{\text{i}}} \\ { - 5}&{\text{i}}&{4 - 2{\text{i}}} \end{array}} \right],\] then AA ∗ will be (where, A ∗ is the conjugate transpose of A)”. This question is standard in competitive examination syllabi for Linear Algebra.

-3 is the correct answer to “Consider the following system of linear equations: 3x + 2ky = -2 kx + 6y = 2 Here, x and y are the unknown and k is a real constant. The value of k for which there are infinite number of solutions is”. This question is standard in competitive examination syllabi for Linear Algebra.

1, 4. 3 is the correct answer to “Consider the matrix as given below: \[\left[ {\begin{array}{*{20}{c}} 1&2&3 \\ 0&4&7 \\ 0&0&3 \end{array}} \right]\] Which one of the following options provides the CORRECT values of the eigen values of the matrix”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\frac{{14}}{3}\] is the correct answer to “The two Eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 2&1 \\ 1&{\text{p}} \end{array}} \right]\] have a ratio of 3 : 1 for p = 2. What is another value of p for which the Eigen values have the same ratio of 3 : 1”. This question is standard in competitive examination syllabi for Linear Algebra.

7 is the correct answer to “The following simultaneous equations x + y + z = 3 x + 2y + 3z = 4 x + 4y + kz = 6 will NOT have a unique solution for k equal to”. This question is standard in competitive examination syllabi for Linear Algebra.

[S] is symmetric and [D] is skew-symmetric is the correct answer to “[A] is square matrix which is neither symmetric nor skew-symmetric and [A] T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A] T and [D] = [A] - [A] T , respectively. Which of the following statements is TRUE”. This question is standard in competitive examination syllabi for Linear Algebra.

-6 and 5 is the correct answer to “The Eigen values of the matrix \[\left[ {\text{P}} \right] = \left[ {\begin{array}{*{20}{c}} 4&5 \\ 2&{ - 5} \end{array}} \right]\] are”. This question is standard in competitive examination syllabi for Linear Algebra.

2n real values not necessarily distinct is the correct answer to “If A is square symmetrical real valued matrix of dimensions 2n, then eigen values of A are”. This question is standard in competitive examination syllabi for Linear Algebra.

(0, -1, -3) is the correct answer to “The eigen values of the matrix given below are \[\left[ {\begin{array}{*{20}{c}} 0&1&0 \\ 0&0&1 \\ 0&{ - 3}&{ - 4} \end{array}} \right]\]”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\left[ {\begin{array}{*{20}{c}} { - 6} \\ { - 3} \\ 6 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 4 \\ { - 2} \\ 3 \end{array}} \right]\] is the correct answer to “\[{\text{P}} = {\left[ {\begin{array}{*{20}{c}} { - 10} \\ { - 1} \\ 3 \end{array}} \right]^{\text{T}}},{\text{Q}} = {\left[ {\begin{array}{*{20}{c}} { - 2} \\ { - 5} \\ 9 \end{array}} \right]^{\text{T}}}\] and \[{\text{R}} = {\left[ {\begin{array}{*{20}{c}} 2 \\ { - 7} \\ {12} \end{array}} \right]^{\text{T}}}\] are three vectors. An orthogonal set of vectors having a span that contains P, Q, R is”. This question is standard in competitive examination syllabi for Linear Algebra.

3, -5 is the correct answer to “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”. This question is standard in competitive examination syllabi for Linear Algebra.

I only is the correct answer to “Let A be n × n real valued square symmetric matrix of rank 2 with \[\sum\limits_{{\text{i}} = 1}^{\text{n}} {\sum\limits_{{\text{j}} = 1}^{\text{n}} {{\text{A}}_{{\text{ij}}}^2} } = 50.\] 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”. This question is standard in competitive examination syllabi for Linear Algebra.

Statement 1 is true but 2 is false is the correct answer to “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”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\frac{1}{2}\] is the correct answer to “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) =”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\left[ {\begin{array}{*{20}{c}} 2&4 \\ 3&6 \end{array}} \right]\] is the correct answer to “Which one of the following matrices is singular”. This question is standard in competitive examination syllabi for Linear Algebra.

infinitely many solutions is the correct answer to “Let c 1 ..... c n be scalars, not all zero, such that \[\sum\limits_{{\text{i}} = 1}^{\text{n}} {{{\text{c}}_{\text{i}}}{{\text{a}}_{\text{i}}}} = 0\] where a i are column vectors in R n . Consider the set of linear equations Ax = b where A = [a 1 ..... a n ] and \[{\text{b}} = \sum\limits_{{\text{i}} = 1}^{\text{n}} {{{\text{a}}_{\text{i}}}} .\] The set of equations has”. This question is standard in competitive examination syllabi for Linear Algebra.

N is the correct answer to “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”. This question is standard in competitive examination syllabi for Linear Algebra.

All eigen values of P are real numbers is the correct answer to “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”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\left[ {\begin{array}{*{20}{c}} 6&2 \\ 2&6 \end{array}} \right]\] is the correct answer to “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”. This question is standard in competitive examination syllabi for Linear Algebra.

\[ - \left( {\frac{{{{\text{A}}^{ - 1}}{\text{b}}}}{2}} \right)\] is the correct answer to “A scalar valued function is defined as f(X) = X T AX + b T X + 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”. This question is standard in competitive examination syllabi for Linear Algebra.

0 is the correct answer to “The minimum eigen value of the following matrix is \[\left[ {\begin{array}{*{20}{c}} 3&5&2 \\ 5&{12}&7 \\ 2&7&5 \end{array}} \right]\]”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\frac{3}{2}\] is the correct answer to “With reference to the conventional Cartesian (x, y) coordinate system, the vertices of a triangle have the following coordinates; (x 1 , y 1 ) = (1, 0); (x 2 , y 2 ) = (2, 2); (x 3 , y 3 ) = (4, 3). The area of the triangle is equal to”. This question is standard in competitive examination syllabi for Linear Algebra.

are always real is the correct answer to “If a square matrix A is real and symmetric, then the eigen values”. This question is standard in competitive examination syllabi for Linear Algebra.

[1 2 1]T is the correct answer to “An eigen vector of \[{\text{P}} = \left[ {\begin{array}{*{20}{c}} 1&1&0 \\ 0&2&2 \\ 0&0&3 \end{array}} \right]\] is”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\left[ {\begin{array}{*{20}{c}} 2 \\ { - 1} \end{array}} \right]\] is the correct answer to “Given the matrix \[\left[ {\begin{array}{*{20}{c}} { - 4}&2 \\ 4&3 \end{array}} \right],\] the eigen vector is”. This question is standard in competitive examination syllabi for Linear Algebra.

zero is the correct answer to “At least one eigen value of a singular matrix is”. This question is standard in competitive examination syllabi for Linear Algebra.

real is the correct answer to “Eigen values of a real symmetric matrix are always”. This question is standard in competitive examination syllabi for Linear Algebra.

either zero or pure imaginary is the correct answer to “The eigen values of a skew-symmetric matrix are”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\left| {\begin{array}{*{20}{c}} 1&{{\text{x}}\left( {{\text{x}} + 1} \right)}&{{\text{x}} + 1} \\ 1&{{\text{y}}\left( {{\text{y}} + 1} \right)}&{{\text{y}} + 1} \\ 1&{{\text{z}}\left( {{\text{z}} + 1} \right)}&{{\text{z}} + 1} \end{array}} \right|\] is the correct answer to “Which one of the following does NOT equal \[\left| {\begin{array}{*{20}{c}} 1&{\text{x}}&{{{\text{x}}^2}} \\ 1&{\text{y}}&{{{\text{y}}^2}} \\ 1&{\text{z}}&{{{\text{z}}^2}} \end{array}} \right|?\]”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\frac{1}{3}\left[ {\begin{array}{*{20}{c}} { - 7}&2 \\ 5&{ - 1} \end{array}} \right]\] is the correct answer to “The inverse of the 2 × 2 matrix \[\left[ {\begin{array}{*{20}{c}} 1&2 \\ 5&7 \end{array}} \right]\] is”. This question is standard in competitive examination syllabi for Linear Algebra.