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