Engineering Mathematics Matrices MCQ

16. Given the matrices $J=\begin{bmatrix} 3 & 2 & 1\\ 2 & 4 & 2\\ 1 & 2 & 6 \end{bmatrix} \; and \; \begin{bmatrix} 1\\ 2\\ -1 \end{bmatrix}$, the product KTJK is

  1. 21
  2. 25
  3. 23
  4. 48
Answer
Answer. c

17. 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 the above statements, which of the following applies?

  1. statements (i) is true but (ii) is false
  2. statement (i) is false but (ii) is true
  3. both the statements are true
  4. both the statements are false
Answer
Answer. a

18. Let A be an m x n matrix and B an n x m matrix. It is given that determinant (Im + AB) = determinant (In + BA), where Ik is the k x k identity matrix. Using the above property, the determinant of the matrix given below is

\[\begin{bmatrix} 2 & 1 & 1 & 1\\ 1 & 2 & 1 & 1\\ 1 & 1 & 2 & 1\\ 1 & 1 & 1 & 2 \end{bmatrix}\]

  1. 2
  2. 5
  3. 8
  4. 16
Answer
Answer. b

19. There are three matrices P(4 x 2), Q(2 x 4) and R(4 x 1). The minimum of multiplication required to compute the matrix PQR is

  1. 12
  2. 15
  3. 18
  4. 16
Answer
Answer. d

20. Given that

\[A=\begin{bmatrix} -5 & -3\\ 2 & 0 \end{bmatrix} \; and \; I=\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\]

the value of A3 is

  1. 15A + 12I
  2. 19A + 30I
  3. 17A + 15I
  4. 17A + 21I
Answer
Answer. b

21. The inverse of the matrix $\begin{bmatrix} 3+2i & i\\ -i & 3-2i \end{bmatrix}$ is

  1. $\frac{1}{12}\begin{bmatrix} 3+2i & -i\\ i & 3-2i \end{bmatrix}$
  2. $\frac{1}{12}\begin{bmatrix} 3-2i & -i\\ i & 3+2i \end{bmatrix}$
  3. $\frac{1}{14}\begin{bmatrix} 3+2i & -i\\ i & 3-2i \end{bmatrix}$
  4. $\frac{1}{14}\begin{bmatrix} 3-2i & -i\\ i & 3+2i \end{bmatrix}$
Answer
Answer. b

22. For a matrix $M=\begin{bmatrix} \frac{3}{5} & \frac{4}{5}\\ x & \frac{3}{5} \end{bmatrix}$, the transpose of the matrix is equal to the inverse of the matrix, [M]T = [M]-1. The value of x is given by

  1. $-\frac{4}{5}$
  2. $-\frac{3}{5}$
  3. $\frac{3}{5}$
  4. $\frac{4}{5}$
Answer
Answer. a

23. A square matrix B is skew-symmetric if

  1. BT = -B
  2. BT = B
  3. B-1 = B
  4. B-1 = BT
Answer
Answer. a

24. The characteristic equation of a (3 x 3) matrix P is defined as

a(λ) = |P – λI| = λ3 + λ2 + 2λ + 1 = 0

If I denotes identity matrix, then the inverse of matrix P will be

  1. P2 + P + 2I
  2. P2 + P + I
  3. -(P2 + P + I)
  4. -(P2 + P + 2I)
Answer
Answer. d

25. The inverse of the 2 x 2 matrix $\begin{bmatrix} 1 & 2\\ 5 & 7 \end{bmatrix}$ is

  1. $\frac{1}{3}\begin{bmatrix} -7 & 2\\ 5 & -1 \end{bmatrix}$
  2. $\frac{1}{3}\begin{bmatrix} 7 & 2\\ 5 & 1 \end{bmatrix}$
  3. $\frac{1}{3}\begin{bmatrix} 7 & -2\\ -5 & 1 \end{bmatrix}$
  4. $\frac{1}{3}\begin{bmatrix} -7 & -2\\ -5 & -1 \end{bmatrix}$
Answer
Answer. a

26. [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?

  1. both [S] and [D] are symmetric
  2. both [S] and [D] are skew symmetric
  3. [S] is skew symmetric and [D] is symmetric
  4. [S] is symmetric and [D] is skew symmetric
Answer
Answer. d

27. Let $A=\begin{bmatrix} 2 & -0.1\\ 0 & 3 \end{bmatrix} \; and \; A^{-1}\begin{bmatrix} \frac{1}{2} & a\\ 0 & b \end{bmatrix}$. Then (a+b) =

  1. $\frac{7}{20}$
  2. $\frac{3}{20}$
  3. $\frac{19}{60}$
  4. $\frac{11}{20}$
Answer
Answer. a

28. Let $R=\begin{bmatrix} 1 & 0 & -1\\ 2 & 1 & -1\\ 2 & 3 & 2 \end{bmatrix}$, Then top row of R-1 is

  1. [5  6  4]
  2. [5  -3  1]
  3. [2  0  -1]
  4. [2  -1   1/2]
Answer
Answer. b

29. Consider the matrices X(4 x 3), Y(4 x 3) and P(2 x 3). The order of [P(XTY)-1PT]T will be

  1. (2 x 2)
  2. (3 x 3)
  3. (4 x 3)
  4. (3 x 4)
Answer
Answer. a

30. Let A, B, C, D be n x n matrices, each with non-zero determinant, if ABCD = I, then B-1 is

  1. D-1C-1A-1
  2. CDA
  3. ADC
  4. does not necessarily exist
Answer
Answer. b

31. For which value of x will the matrix given below becomes singular?

\[\begin{bmatrix} 8 & x & 0\\ 4 & 0 & 2\\ 12 & 6 & 0 \end{bmatrix}\]

  1. 4
  2. 6
  3. 8
  4. 12
Answer
Answer. a

32. The product of matrices (PQ)-1P is

  1. P-1
  2. Q-1
  3. P-1Q-1P
  4. PQP-1
Answer
Answer. b
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