Matrices: Types, Equality, Transpose, Algebra of Matrices

If \(A = [a_{ij}]\) is a \(3 \times 3\) matrix with \(a_{ij} = 2i - j\), then \(a_{23}\) equals:

1. a4
2. b1
3. c7
4. d\(-1\)

A matrix has 12 elements. How many different orders can it have?

1. a3
2. b4
3. c6
4. d12

Match each matrix in List I with its type in List II:

1. a(i)-R, (ii)-S, (iii)-P, (iv)-Q
2. b(i)-R, (ii)-P, (iii)-S, (iv)-Q
3. c(i)-Q, (ii)-S, (iii)-P, (iv)-R
4. d(i)-Q, (ii)-P, (iii)-S, (iv)-R

1. I. Every identity matrix is a scalar matrix.
2. II. Every scalar matrix is an identity matrix.
3. III. A zero matrix must be a square matrix.

Which of the statements above about types of matrices is/are correct?

1. aOnly I and II
2. bOnly I
3. cOnly I and III
4. dOnly II

Assertion (A): The matrix \(\begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}\) is a diagonal matrix.

Reason (R): A square matrix in which every element off the main diagonal is zero is called a diagonal matrix.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

If \(\begin{bmatrix} x + y & 2 \\ 5 & x - y \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 2 \end{bmatrix}\), then:

1. a\(x = 4,\ y = 2\)
2. b\(x = 2,\ y = 4\)
3. c\(x = 3,\ y = 3\)
4. d\(x = 4,\ y = -2\)

If \(A = \begin{bmatrix} 1 & -2 & 3 \\ 0 & 4 & 5 \end{bmatrix}\), then \(A'\) is:

1. a\(\begin{bmatrix} 1 & -2 \\ 3 & 0 \\ 4 & 5 \end{bmatrix}\)
2. b\(\begin{bmatrix} 0 & 4 & 5 \\ 1 & -2 & 3 \end{bmatrix}\)
3. c\(\begin{bmatrix} 1 & 0 \\ -2 & 4 \\ 3 & 5 \end{bmatrix}\)
4. d\(\begin{bmatrix} 3 & 5 \\ -2 & 4 \\ 1 & 0 \end{bmatrix}\)

1. I. \((A')' = A\)
2. II. \((A + B)' = A' + B'\)
3. III. \((AB)' = A'B'\)

Here \(A\) and \(B\) are matrices for which the sum and product are defined. Which of the statements above about transposes is/are correct?

1. aOnly I
2. bOnly II and III
3. cI, II and III
4. dOnly I and II

Assertion (A): Every diagonal element of a skew-symmetric matrix is zero.

Reason (R): In a skew-symmetric matrix \(M = [m_{ij}]\), \(m_{ji} = -m_{ij}\) for all \(i\) and \(j\); in particular \(m_{ii} = -m_{ii}\).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

For what value of \(k\) is the matrix \(\begin{bmatrix} 0 & 3 & k - 1 \\ -3 & 0 & 5 \\ 4 & -5 & 0 \end{bmatrix}\) skew-symmetric?

1. a5
2. b\(-3\)
3. c3
4. d\(-4\)

The matrix \(A = \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix}\) is written as \(A = P + Q\), where \(P\) is symmetric and \(Q\) is skew-symmetric. Then \(Q\) is:

1. a\(\begin{bmatrix} 0 & 4 \\ -4 & 0 \end{bmatrix}\)
2. b\(\begin{bmatrix} 3 & 3 \\ 3 & 2 \end{bmatrix}\)
3. c\(\begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix}\)
4. d\(\begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}\)

Assertion (A): If \(X\) and \(Y\) are symmetric matrices of the same order, then \(X + Y\) is symmetric.

Reason (R): If \(X\) and \(Y\) are symmetric matrices of the same order, then \(XY\) is always symmetric.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

If \(A = \begin{bmatrix} 2 & -1 \\ 3 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 4 \\ -2 & 5 \end{bmatrix}\), then \(2A - B\) is:

1. a\(\begin{bmatrix} 3 & -6 \\ 8 & -5 \end{bmatrix}\)
2. b\(\begin{bmatrix} 5 & 2 \\ 4 & 5 \end{bmatrix}\)
3. c\(\begin{bmatrix} 1 & -5 \\ 5 & -5 \end{bmatrix}\)
4. d\(\begin{bmatrix} 3 & -6 \\ 4 & -5 \end{bmatrix}\)

A uniform shop keeps its stock in matrices with rows (shirts, trousers) and columns (small, large). Its stock on Monday morning is \(S = \begin{bmatrix} 50 & 30 \\ 40 & 20 \end{bmatrix}\). During the week it sells \(T = \begin{bmatrix} 35 & 12 \\ 18 & 20 \end{bmatrix}\), and on Saturday it receives a delivery \(D = \begin{bmatrix} 10 & 15 \\ 20 & 25 \end{bmatrix}\). Its stock at the end of the week is:

1. a\(\begin{bmatrix} 5 & 3 \\ 2 & -25 \end{bmatrix}\)
2. b\(\begin{bmatrix} 25 & 33 \\ 42 & 25 \end{bmatrix}\)
3. c\(\begin{bmatrix} 75 & 27 \\ 38 & 15 \end{bmatrix}\)
4. d\(\begin{bmatrix} 95 & 57 \\ 78 & 65 \end{bmatrix}\)

If \(A\) is a \(2 \times 3\) matrix and \(B\) is a \(3 \times 4\) matrix, then the order of \(AB\) is:

1. a\(3 \times 3\)
2. b\(4 \times 2\)
3. c\(2 \times 3\)
4. d\(2 \times 4\)

If \(A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}\), then \(AB\) is:

1. a\(\begin{bmatrix} 2 & 4 \\ 10 & 14 \end{bmatrix}\)
2. b\(\begin{bmatrix} 2 & 0 \\ 3 & 12 \end{bmatrix}\)
3. c\(\begin{bmatrix} 4 & 6 \\ 10 & 12 \end{bmatrix}\)
4. d\(\begin{bmatrix} 4 & 10 \\ 6 & 12 \end{bmatrix}\)

For \(A = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}\), match List I with List II:

1. a(i)-Q, (ii)-R, (iii)-P, (iv)-S
2. b(i)-Q, (ii)-R, (iii)-S, (iv)-P
3. c(i)-R, (ii)-Q, (iii)-S, (iv)-P
4. d(i)-Q, (ii)-S, (iii)-R, (iv)-P

1. I. If the product \(AB\) is defined, then \(BA\) must also be defined.
2. II. There exist non-zero matrices \(A\) and \(B\) such that \(AB = O\).
3. III. For square matrices \(A\) and \(B\) of the same order, \((A + B)^{2} = A^{2} + 2AB + B^{2}\) always.

Which of the statements above about matrix multiplication is/are correct?

1. aOnly II
2. bOnly I and II
3. cOnly II and III
4. dI, II and III

If \(A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix}\) and \(A^{2} = kA - 2I\), where \(I\) is the identity matrix of order 2, then \(k\) equals:

1. a\(-1\)
2. b4
3. c1
4. d\(-2\)

A carpentry workshop makes chairs and tables. The units of wood, nails and paint needed are given by \(R = \begin{bmatrix} 2 & 1 & 3 \\ 1 & 2 & 2 \end{bmatrix}\) (row 1: one chair, row 2: one table; columns: wood, nails, paint). One unit of wood, nails and paint costs ₹ 50, ₹ 100 and ₹ 40 respectively. Using matrix multiplication, what is the total material cost of an order for 20 chairs and 30 tables?

1. a₹ 16,200
2. b₹ 16,800
3. c₹ 650
4. d₹ 16,300