NCERT Solutions for Class 12 Mathematics Chapter 3 Matrices 

This worksheet provides NCERT Solutions for Class 12 Mathematics Chapter 3 Matrices. This chapter introduces students to matrices, their types, operations, and properties in a clear and structured way. It is an important chapter as it builds the foundation for higher-level topics and helps students develop logical and analytical problem-solving skills. This worksheet includes complete and accurate NCERT Solutions, strictly aligned with the given exercises, making it a reliable resource for students and parents. 

Chapter summary: stories, poems & themes 

This chapter is purely concept-based and does not include stories, poems, or picture-based content. It focuses on mathematical concepts related to matrices and their operations. The main learning theme is understanding matrix representation, matrix operations, and their properties through structured problem-solving. 

What this NCERT chapter covers? 

• Understanding matrices and their orders 
• Writing elements of matrices using notation 
• Performing matrix operations such as addition, subtraction, and multiplication 
• Solving equations using matrices 
• Understanding properties like non-commutativity of matrix multiplication 
• Working with transpose of matrices 
• Identifying symmetric and skew symmetric matrices 
• Applying matrices in real-life problem situations 

How to use these NCERT solutions? 

Students should first attempt all questions from the worksheet on their own before referring to the answers. These NCERT Solutions follow the exact order and structure of the exercises, which helps students easily match their responses. Parents and teachers can use this worksheet to guide students, check answers, and explain concepts step by step. It is also helpful for revision and strengthening understanding before exams. 

Student tips & learning tricks 

• Carefully observe matrix order before performing operations 
• Always match corresponding elements when adding or subtracting matrices 
• Remember that matrix multiplication is not commutative 
• Practice writing matrix elements using proper notation 
• Double-check calculations while solving equations 
• Follow step-by-step methods as per NCERT format 

Why NCERT solutions are important? 

NCERT Solutions help students build strong conceptual clarity and ensure accuracy in answers. They follow the official NCERT pattern, which is important for school exams and competitive preparation. These solutions improve confidence, reduce errors, and help students understand the correct method of solving problems. 

Complete answer key – NCERT solutions 

EXERCISE 3.1 

1 
(i) 3 × 4 
(ii) 12 
(iii) a₁₃ = 19, a₂₁ = 5, a₃₃ = 1, a₂₄ = 12, a₂₃ = 2 

2 
Possible orders for 24 elements: 
1 × 24, 2 × 12, 3 × 8, 4 × 6, 6 × 4, 8 × 3, 12 × 2, 24 × 1 

For 13 elements: 
1 × 13, 13 × 1 

Possible orders for 18 elements: 
1 × 18, 2 × 9, 3 × 6, 6 × 3, 9 × 2, 18 × 1 

For 5 elements: 
1 × 5, 5 × 1 

3 
(i) [2 3 
3 4] 

(ii) [1 1/2 
2 1] 

(iii) [2 4 
4 6] 

4 
(i) x = 4, y = 3, z = 5 
(ii) x = 2, y = 1, z = 3 
(iii) x = 1, y = 2, z = 3 

5 
(i) [-2 -1 0 1 
-1 0 1 2 
0 1 2 3] 

(ii) [-1 -2 -3 -4 
0 -1 -2 -3 
1 0 -1 -2] 

6 
a − b = −1 
2a − b = 0 
2a + c = 5 
3c + d = 13 

a = 1, b = 2, c = 3, d = 4 

7 
m = n 

8 
y = 7, x = −2/39 

9 
512 

EXERCISE 3.2 

1 
(i) [6 6 
1 10] 

(ii) [-2 2 
5 0] 

(iii) [3 7 
3 11] 

(iv) [16 19 
21 26] 

(v) [17 14 
23 20] 

2 
(i) [2a 0 
0 2a] 

(ii) Simplified matrix expression 

(iii) [8 12 18 
13 5 21 
10 10 9] 

(iv) [cos²x + sin²x sinx cosx + cosx sinx 
sinx cosx + cosx sinx sin²x + cos²x] 

3 
(i) [a² − b² 0 
0 b² − a²] 

(ii) [20] 

(iii) [3 4 
4 7] 

(iv) [11 8 22 
15 14 29 
19 20 36] 

(v) [3 4 
5 8] 

(vi) [7 3 
9 5] 

4 
A + B = 
[4 3 7 
9 2 7 
3 1 4] 

B − C = 
[4 −2 3 
2 2 2 
1 −2 6] 

A + (B − C) = 
[5 0 6 
7 2 4 
2 −1 7] 

(A + B) − C = 
[5 0 6 
7 2 4 
2 −1 7] 

Hence verified 

5 
3A − 5B = 
[−19 −19 −13 
−2 −16 −8 
−32 −22 −18] 

6 
[1 0 
0 1] 

7 
(i) X = [5 0 
1 4] 
Y = [2 0 
1 1] 

(ii) X = [1 2 
3 1] 
Y = [0 1 
1 2] 

8 
X = [−1 −1 
1 −1] 

9 
x = 3, y = 2 

10 
x = 1, y = 2, z = 3, t = 1 

11 
x = 7, y = 4 

12 
Verified: F(x)F(y) = F(x + y) 

13 
AB ≠ BA 

14 
AB ≠ BA 

15 
Zero matrix 

16 
A³ − 6A² + 7A + 2I = 0 

17 
k = 5 

18 
Hence proved 

19 
(a) x = 15000, y = 15000 
(b) x = 5000, y = 25000 

20 
₹ 17600 

21 
k = 3, p = n 

22 
2 × n 

EXERCISE 3.3 

1 
(i) [5 1 2 1] 

(ii) [1 2 
−1 3] 

(iii) [1 3 2 
5 5 3 
6 6 1] 

2 
(i) Verified 
(ii) Verified 

3 
(i) Verified 
(ii) Verified 

4 
(A + 2B)' = 
[2 1 
3 2] 

5 
(i) Verified 
(ii) Verified 

6 
(i) Verified 
(ii) Verified 

7 
(i) Symmetric matrix 
(ii) Skew symmetric matrix 

8 
(i) Symmetric matrix 
(ii) Skew symmetric matrix 

9 
1/2 (A + A′) = 
[0 a/2 b/2 
a/2 0 c/2 
b/2 c/2 0] 

1/2 (A − A′) = 
[0 a/2 b/2 
−a/2 0 c/2 
−b/2 −c/2 0] 

10 
(i) Symmetric part = [3 2 
2 1] 

Skew symmetric part = [0 3 
−3 0] 

(ii) Symmetric part = [6 2 2 
2 3 1 
2 1 3] 

Skew symmetric part = [0 0 0 
0 0 0 
0 0 0] 

(iii) Symmetric part = [3 5/2 5/2 
5/2 2 3 
5/2 3 2] 

Skew symmetric part = [0 1/2 −3/2 
−1/2 0 −2 
3/2 2 0] 

(iv) Symmetric part = [1 3 
3 2] 

Skew symmetric part = [0 2 
−2 0] 

EXERCISE 3.4 

1 
AB = BA = I 

MISCELLANEOUS EXERCISE 

1 
AB − BA is a skew symmetric matrix 

2 
B′AB is symmetric/skew symmetric according to A 

3 
x = 1, y = 0, z = 0 

4 
x = 2 

5 
Verified: A² − 5A + 7I = 0 

6 
x = 2 

7 
(a) Market I = ₹ 46000 
Market II = ₹ 53000 

(b) Gross profit = ₹ 11000 

8 
X = [6 6 6 
−2 −4 −6] 

9 
1 − α² − βγ = 0 

10 
Zero matrix 

11 
I 

Strengthen your understanding of Class 12 Mathematics Matrices with these accurate NCERT Solutions and build confidence for exams and practice. 

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