NCERT Solutions for Class 12 Maths Chapter 3

NCERT Solutions for Class 12 Maths Chapter 3 Matrices

NCERT

Exercise 3.1

Question 1:
In the matrix

$$A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12\\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix},$$ write:
(i) The order of the matrix, (ii) The number of elements, (iii) Write the elements a₁₃, a₂₁, a₃₃, a₂₄, a₂₃.
Answer
(i) Order of the matrix is 3 x 4.
(ii) The number of elements in the matrix A is 3 x 4 = 12.
(iii) a ₁₃ = 19; a ₂₁ = 35; a ₃₃ = -5; a ₂₄ = 12; a ₂₃ = $\frac{5}{2}$

Question 2:
If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?
Answer
We know that if a matrix is of the order m × n, it has mn elements.
The ordered pairs are: (1, 24), (24, 1), (2, 12), (12, 2), (3, 8), (8, 3), (4, 6), and (6, 4)
Hence, the possible orders of a matrix having 24 elements are: 1 × 24, 24 × 1, 2 × 12, 12 × 2, 3 × 8, 8 × 3, 4 × 6, and 6 × 4
(1, 13) and (13, 1) are the ordered pairs of natural numbers whose product is 13.
Hence, the possible orders of a matrix having 13 elements are 1 × 13 and 13 × 1.

Question 3:
If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
Answer
We know that if a matrix is of the order m × n, it has mn elements.
The ordered pairs are: (1, 18), (18, 1), (2, 9), (9, 2), (3, 6,), and (6, 3)
Hence, the possible orders of a matrix having 18 elements are: 1 × 18, 18 × 1, 2 × 9, 9 × 2, 3 × 6, and 6 × 3
(1, 5) and (5, 1) are the ordered pairs of natural numbers whose product is 5.
Hence, the possible orders of a matrix having 5 elements are 1 × 5 and 5 × 1.

Question 4:
Construct a 2 × 2 matrix, A = [a_ij], whose elements are given by:
$(i) \; a_{ij} = \frac{(i + j)^2}{2} \quad (ii) \; a_{ij} = \frac{i}{j} \quad (iii) \; a_{ij} = \frac{(i + 2j)^2}{2}$
Answer
$(i) \; a_{ij} = \frac{(i + j)^2}{2}$
$\quad$ a₁₁ = $\frac{(1 + 1)^2}{2}$ = 2; $\quad$ a₁₂ = $\frac{(1 + 2)^2}{2}$ = $\frac{9}{2}$
$\quad$ a₂₁ = $\frac{(2 + 1)^2}{2}$ = $\frac{9}{2}$; $\quad$ a₂₂ = $\frac{(2 + 2)^2}{2}$ = 8
$\quad$ $A = \begin{bmatrix} 2 & \frac{9}{2} \\ \frac{9}{2} & 8\\ \end{bmatrix}$

$(ii) \; a_{ij} = \frac{i}{j}$
$\quad$ a₁₁ = $\frac{1}{1}$ = 1; $\quad$ a₁₂ = $\frac{1}{2}$;
$\quad$ a₂₁ = $\frac{2}{1}$ = 2; $\quad$ a₂₂ = $\frac{2}{2}$ = 1
$\quad$ $A = \begin{bmatrix} 1 & \frac{1}{2} \\ 2 & 1\\ \end{bmatrix}$

$(iii) \; a_{ij} = \frac{(i + 2j)^2}{2}$
$\quad$ a₁₁ = $\frac{(1 + 2(1))^2}{2}$ = $\frac{9}{2}$; $\quad$ a₁₂ = $\frac{(1 + 2(2))^2}{2}$ = $\frac{25}{2}$;
$\quad$ a₂₁ = $\frac{(2 + 2(1))^2}{2}$ = 8; $\quad$ a₂₂ = $\frac{(2 + 2(2))^2}{2}$ = 18
$\quad$ $A = \begin{bmatrix} \frac{9}{2} & \frac{25}{2} \\ 8 & 18\\ \end{bmatrix}$

Question 5:
Construct a 3 × 4 matrix, whose elements are given by:
$(i)\; a_{ij} = \frac{1}{2} |-3i + j| \quad (ii) \; a_{ij} = 2i – j$
Answer
$(i)\; a_{ij} = \frac{1}{2} |-3i + j|$
$\quad$ a₁₁ = $\frac{1}{2} |-3(1) + 1|$ = 1; $\quad$ a₁₂ = $\frac{1}{2} |-3(1) + 2)|$ = $\frac{1}{2}$; $\quad$ a₁₃ = $\frac{1}{2} |-3(1) + 3)|$ = 0; $\quad$ a₁₄ = $\frac{1}{2} |-3(1) + 4|$ = $\frac{1}{2}$;
$\quad$ a₂₁ = $\frac{1}{2} |-3(2) + 1)|$ = $\frac{5}{2}$; $\quad$ a₂₂ = $\frac{1}{2} |-3(2) + 2)|$ = 2; $\quad$ a₂₃ = $\frac{1}{2} |-3(2) + 3)|$ = $\frac{3}{2}$; $\quad$ a₂₄ = $\frac{1}{2} |-3(2) + 4|$ = 1;
$\quad$ a₃₁ = $\frac{1}{2} |-3(3) + (1)|$ = 4; $\quad$ a₃₂ = $\frac{1}{2} |-3(3) + 2)|$ = $\frac{7}{2}$; $\quad$ a₃₃ = $\frac{1}{2} |-3(3) + 3)|$ = 3; $\quad$ a₃₄ = $\frac{1}{2} |-3(3) + 4|$ = $\frac{5}{2}$;
$\quad$ $A = \begin{bmatrix} 1 & \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{5}{2} & 2 & \frac{3}{2} & 1\\ 4 & \frac{7}{2} & 3 & \frac{5}{2} \end{bmatrix}$

$(ii) \; a_{ij} = 2i – j$
$\quad$

Question 6:
Find the values of x, y and z from the following equations:

$$(i)\; \begin {bmatrix} 4 & 3 \\ x & 5 \end {bmatrix} = \begin {bmatrix} y & z \\ 1 & 5 \end {bmatrix} (ii)\; \begin {bmatrix} x + y & 2 \\ 5 + z & xy \end {bmatrix} = \begin {bmatrix} 6 & 2 \\ 5 & 5 \end {bmatrix} (iii)\; \begin {bmatrix} x + y + z \\ x + z \\ y + z \\ \end {bmatrix} = \begin {bmatrix} 9 \\ 5 \\ 7 \end {bmatrix}$$

Question 7:
Find the value of a, b, c and d from the equation:

$$\begin {bmatrix} a-b & 2a + c \\ 2a – b & 3c + d \end {bmatrix} = \begin {bmatrix} -1 & 5 \\ 0 & 13 \end {bmatrix}$$

Question 8:
A = [a_ij]_{m × n} is a square matrix, if
(A) m < n (B) m > n (C) m = n (D) None of these

Question 9:
Which of the given values of x and y make the following pair of matrices equal

$$\begin {bmatrix} 3x + 7 & 5 \\ y + 1 & 2 – 3x \end {bmatrix}, \begin {bmatrix} 0 & y – 2 \\ 8 & 4 \end {bmatrix}$$
$(A)\; x =\frac{-1}{3} \quad (B) \;$ Not possible to find $\quad (C)\; y = 7, x = \frac{-2}{3} \quad (D) \;x = \frac{-1}{3}, y = \frac{-2}{3}$

Question 10:
The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:
(A) 27 (B) 18 (C) 81 (D) 512

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Exercise 3.2

Question 1:
Let $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}, B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}, C = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}$
Find each of the following:
(i) A + B (ii) A – B (iii) 3A – C (iv) AB (v) BA