Linear Algebra

[amp_mcq option1=”Both [S] and [D] are symmetric” option2=”Both [S] and [D] are skew-symmetric” option3=”[S] is skew-symmetric and [D] is symmetric” option4=”[S] is symmetric and [D] is skew-symmetric” correct=”option1″]

Detailed Solution[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? A. Both [S] and [D] are symmetric B. Both [S] and [D] are skew-symmetric C. [S] is skew-symmetric and [D] is symmetric D. [S] is symmetric and [D] is skew-symmetric

[amp_mcq option1=”1, 4. 3″ option2=”3, 7, 3″ option3=”7, 3, 2″ option4=”1, 2, 3″ correct=”option3″]

Detailed SolutionConsider 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? A. 1, 4. 3 B. 3, 7, 3 C. 7, 3, 2 D. 1, 2, 3

[amp_mcq option1=”3″ option2=”1″ option3=”-3″ option4=”-6″ correct=”option3″]

Detailed SolutionConsider 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 A. 3 B. 1 C. -3 D. -6

[amp_mcq option1=”4″ option2=”3″ option3=”2″ option4=”1″ correct=”option1″]

Detailed SolutionGiven 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 A. 4 B. 3 C. 2 D. 1

[amp_mcq option1=”zero” option2=”1″ option3=”2″ option4=”Infinite” correct=”option1″]

Detailed SolutionThe number of solutions of the simultaneous algebraic equation y = 3x + 3 and y = 3x + 5 is: A. zero B. 1 C. 2 D. Infinite

[amp_mcq option1=”4″ option2=”6″ option3=”8″ option4=”12″ correct=”option3″]

Detailed SolutionFor which value of x will the matrix given below become singular? \[\left[ {\begin{array}{*{20}{c}} 8&{\text{x}}&0 \\ 4&0&2 \\ {12}&6&0 \end{array}} \right]\] A. 4 B. 6 C. 8 D. 12

[amp_mcq option1=”1″ option2=”2″ option3=”3″ option4=”4″ correct=”option1″]

Detailed SolutionThe rank of the matrix \[\left[ {\begin{array}{*{20}{c}} { – 4}&1&{ – 1} \\ { – 1}&{ – 1}&{ – 1} \\ 7&{ – 3}&1 \end{array}} \right]\] is A. 1 B. 2 C. 3 D. 4

[amp_mcq option1=”7a – b – c = 0″ option2=”3a + b – c = 0″ option3=”3a – b + c = 0″ option4=”7a – b + c = 0″ correct=”option1″]

Detailed SolutionConsider 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 A. 7a – b – c = 0 B. 3a + b – c = 0 C. 3a – b + c = 0 D. 7a – b + c = 0

[amp_mcq option1=”(2 × 2)” option2=”(3 × 3)” option3=”(4 × 3)” option4=”(3 × 4)” correct=”option1″]

Detailed SolutionConsider the matrices X(4 × 3), Y(4 × 3) and P(2 × 3). The order of [P(XTY)-1 PT]T will be A. (2 × 2) B. (3 × 3) C. (4 × 3) D. (3 × 4)

[amp_mcq option1=”BT = -B” option2=”BT = B” option3=”B-1 = B” option4=”B-1 = BT” correct=”option1″]

Detailed SolutionA square matrix B is skew-symmetric if A. BT = -B B. BT = B C. B-1 = B D. B-1 = BT