NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Matrices | LearnCBSEHub

Matrix Universe

LearnCBSEHub.in • Exercise 3.1 Solutions

💡 Key Concepts & Exam Strategy

🔸 Order ($m \times n$): Total horizontal rows ($m$) and vertical columns ($n$).
🔸 Addressing: $a_{ij}$ means element at $i^{th}$ row and $j^{th}$ column.
🔸 Square Matrix: A matrix where $m = n$.
🔸 Construction Tip: For $a_{ij}$ formulas, solve for each $(i, j)$ pair methodically.

Question 01

For matrix $A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & 5/2 & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$, find Order, Elements count, and values.

(i) Order: 3 Rows $\times$ 4 Columns $\rightarrow$ $3 \times 4$

(ii) Total Elements: $3 \times 4 = \mathbf{12}$

(iii) Elements: $a_{13}=19, a_{21}=35, a_{33}=-5, a_{24}=12, a_{23}=5/2$

Question 02 & 03

Find possible orders for a matrix with 24, 13, 18, and 5 elements.

24 elements: (1,24), (24,1), (2,12), (12,2), (3,8), (8,3), (4,6), (6,4)

13 elements: (1,13), (13,1)

18 elements: (1,18), (18,1), (2,9), (9,2), (3,6), (6,3)

5 elements: (1,5), (5,1)

Question 04 & 05

Matrix Construction using $a_{ij}$ formulas.

Q4(i) $2 \times 2$: $a_{ij} = \frac{(i+j)^2}{2} \rightarrow \begin{bmatrix} 2 & 9/2 \\ 9/2 & 8 \end{bmatrix}$

Q5(i) $3 \times 4$: $a_{ij} = \frac{1}{2}|-3i + j|$

$$\begin{bmatrix} 1 & 1/2 & 0 & 1/2 \\ 5/2 & 2 & 3/2 & 1 \\ 4 & 7/2 & 3 & 5/2 \end{bmatrix}$$

Question 06 & 07

Find unknown variables from matrix equations.

Q6(ii): $x+y=6, xy=8, z+5=5 \implies \mathbf{z=0, x=4, y=2}$ or $\mathbf{x=2, y=4}$

Q7: $\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix} \implies \mathbf{a=1, b=2, c=3, d=4}$

Questions 08 — 10

Multiple Choice Solutions.

Q8 Square Matrix: Condition is $m=n$. Option (C)

Q9 Equality: $x$ has conflicting values. Option (B) Not possible

Q10 Possible Matrices: $2^9 = 512$. Option (D) 512