Exercise-3.1, Class 12th, Maths, Chapter 3- Matrices, NCERT

Exercise 3.1

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Question 1:

In the matrix

$$A=\left[\begin{array}{cccc}2& 5& 19& -7\\ 1& -5& 17& 3\\ 1& -2& 12& 5\end{array}\right]$$

write:
(i) The order of the matrix
(ii) The number of elements
(iii) Write the elements a₁₃, a₂₁, a₃₃, a₂₄, a₂₃.

Answer:
(i) Order = 3 × 4
(ii) Number of elements = 3 × 4 = 12
(iii) a₁₃ = 19, a₂₁ = 1, a₃₃ = 12, a₂₄ = 3, a₂₃ = 17.

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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 has m × n elements, then total elements = m × n.

(i) For 24 elements:
Possible orders = (1,24), (2,12), (3,8), (4,6), (6,4), (8,3), (12,2), (24,1)

(ii) For 13 elements (prime number):
Possible orders = (1,13), (13,1)

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Question 3:

If a matrix has 18 elements, what are its possible orders? What if it has 5 elements?

Answer:
(i) For 18 elements: $m\times n=18$
Possible orders = (1,18), (2,9), (3,6), (6,3), (9,2), (18,1)

(ii) For 5 elements (prime): (1,5), (5,1)

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Question 4:

Construct a 2 × 2 matrix A = [aᵢⱼ] where:

(i) aᵢⱼ = 2i − j
(ii) aᵢⱼ = i² − 3j
(iii) aᵢⱼ = i² / 2j

Answer:
For i, j = 1, 2

(i)

$$A=\left[\begin{array}{cc}2(1)-1& 2(1)-2\\ 2(2)-1& 2(2)-2\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 3& 2\end{array}\right]$$

(ii)

$$A=\left[\begin{array}{cc}{1}^2-3(1)& 1^2-3(2)\\ 2^2-3(1)& 2^2-3(2)\end{array}\right]=\left[\begin{array}{cc}-2& -5\\ 1& -2\end{array}\right]$$

(iii)

$$A=\left[\begin{array}{cc}\frac{1^2}{2\times 1}& \frac{1^2}{2\times 2}\\ \frac{2^2}{2\times 1}& \frac{2^2}{2\times 2}\end{array}\right]=\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{4}\\ 2& 1\end{array}\right]$$

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Question 5:

Construct a 3 × 4 matrix where:
(i) aᵢⱼ = ½ (i − 3j)
(ii) aᵢⱼ = 2i − j

Answer:

(i)

$$A=\left[\begin{array}{cccc}-1& -2.5& -4& -5.5\\ -0.5& -2& -3.5& -5\\ 0& -1.5& -3& -4.5\end{array}\right]$$

(ii)

$$A=\left[\begin{array}{cccc}1& 0& -1& -2\\ 3& 2& 1& 0\\ 5& 4& 3& 2\end{array}\right]$$

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Question 6:

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

(i)

$$\left[\begin{array}{ccc}x& 5& z\\ y& 1& 5\end{array}\right]=\left[\begin{array}{ccc}2& 6& 2\\ 1& 5& 8\end{array}\right]$$

(ii)

$$\left[\begin{array}{ccc}5& z& x+y\\ 5& 8& x+y\end{array}\right]=\left[\begin{array}{ccc}x+z& y& z\\ xz& y& z\end{array}\right]$$

Answer:

(i) Comparing elements:
x = 2, y = 1, z = 2.

(ii) On comparing:
x = 3, y = 1, z = 2.

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Question 7:

Find a, b, c, d from the equation:

$$\left[\begin{array}{cc}a-b& 2a-b\\ 2a+c& 3c-d\end{array}\right]=\left[\begin{array}{cc}0& -15\\ 0& 13\end{array}\right]$$

Answer:
Equating elements:
a − b = 0
2a − b = −15
2a + c = 0
3c − d = 13

Solving, we get
a = −15, b = −15, c = 30, d = 77.

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Question 8:

A = [aᵢⱼ] is a square matrix if:
(A) m < n  (B) m > n  (C) m = n  (D) None of these

Answer:
(C) m = n

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Question 9:

Which values of x and y make the following matrices equal?

$$\left[\begin{array}{cc}3x+7& 5\\ y+1& 2\end{array}\right]=\left[\begin{array}{cc}-3x& 0\\ -2& 8\end{array}\right]$$

Answer:
3x + 7 = −3x ⇒ 6x = −7 ⇒ x = −7/6
y + 1 = −2 ⇒ y = −3

x = −7/6, y = −3

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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

Answer:
Each entry can be 0 or 1, i.e., 2 possibilities per element.
Total = $2^{3\times 3}=2^9=512$

Answer: (D) 512