NCERT Solutions Class 12 Mathematics Chapter 3 Matrices

NCERT Solutions Class 12 Mathematics Chapter 3 Matrices offers step-by-step, syllabus-aligned solutions designed to help students master matrix algebra for CBSE Class 12 Board Examinations, CUET, and JEE Main. Matrices form the bedrock of linear algebra and higher mathematics, playing a pivotal role in solving systems of linear equations, computer graphics, cryptography, and engineering data analysis. In this detailed guide, you will find concept breakdowns, verified NCERT solutions step by step, important theorems with proofs, exam weightage insights, and common pitfalls to avoid.

Chapter Overview

Chapter 3 of the Class 12 NCERT Mathematics textbook introduces matrices as rectangular arrangements of numbers or functions. This chapter equips students with algebraic tools to handle multidimensional data arrays systematically. The chapter consists of four primary exercises and a miscellaneous exercise covering:

• Matrix Basics & Types (Exercise 3.1): Definition, order of a matrix (m × n), types of matrices (row, column, zero, square, diagonal, scalar, and identity matrix), and equality of matrices.
• Matrix Operations (Exercise 3.2): Addition, scalar multiplication, negative of a matrix, matrix multiplication, and properties (associative, commutative, distributive laws).
• Transpose of a Matrix (Exercise 3.3): Properties of transpose, symmetric matrices (A^T = A), skew-symmetric matrices (A^T = −A), and expressing any square matrix as the sum of a symmetric and a skew-symmetric matrix.
• Invertibility & Matrix Equations (Exercise 3.4 & Miscellaneous): Invertible matrices, uniqueness of inverse, reversal law (AB)⁻¹ = B⁻¹A⁻¹, and proofs using the Principle of Mathematical Induction (PMI).

In the CBSE Class 12 Mathematics curriculum, the unit Algebra (comprising Chapter 3 Matrices and Chapter 4 Determinants) carries an aggregate weightage of 10 marks in the board theory paper. Matrices contribute typically 4 to 6 marks through 1-mark MCQs, 2-mark or 3-mark short answer questions, and parts of 4-mark case-based questions.

Exercise Solutions: Step-by-Step Whiteboard Solutions

Below are exhaustive, step-by-step solutions for representative and high-frequency questions across all exercises of NCERT Chapter 3 Matrices.

Exercise 3.1: Matrix Order, Element Construction, and Equality

Question 1: Construction of a Matrix

Question: Construct a 3 × 2 matrix A = [a_ij] whose elements are given by a_ij = ½ |−3i + j|.

• Given: Order of matrix A is 3 × 2 (3 rows, 2 columns); formula a_ij = ½ |−3i + j|.
• To Find: The complete matrix A = [[a₁₁, a₁₂], [a₂₁, a₂₂], [a₃₁, a₃₂]].
• Step-by-Step Solution:
  1. For Row 1 (i = 1):
     a₁₁ = ½ |−3(1) + 1| = ½ |−2| = 1
     a₁₂ = ½ |−3(1) + 2| = ½ |−1| = ½
  2. For Row 2 (i = 2):
     a₂₁ = ½ |−3(2) + 1| = ½ |−5| = 5/2
     a₂₂ = ½ |−3(2) + 2| = ½ |−4| = 2
  3. For Row 3 (i = 3):
     a₃₁ = ½ |−3(3) + 1| = ½ |−8| = 4
     a₃₂ = ½ |−3(3) + 2| = ½ |−7| = 7/2
• Final Answer: A = [[1, 1/2], [5/2, 2], [4, 7/2]]

Question 2: Equality of Matrices

Question: Find the values of x, y, z, and w if [[x − y, 2x + z], [2x − y, 3z + w]] = [[−1, 5], [0, 13]].

• Given: Two 2 × 2 matrices are equal.
• Formula / Principle: Two matrices are equal if and only if their corresponding elements are equal (a_ij = b_ij).
• Step-by-Step Solution:
  1. Equate corresponding elements to form a system of linear equations:
     (1) x − y = −1
     (2) 2x − y = 0
     (3) 2x + z = 5
     (4) 3z + w = 13
  2. From equation (2): y = 2x. Substitute into (1): x − 2x = −1 ⇒ −x = −1 ⇒ x = 1.
  3. Substitute x = 1 into y = 2x: y = 2.
  4. Substitute x = 1 into equation (3): 2(1) + z = 5 ⇒ z = 3.
  5. Substitute z = 3 into equation (4): 3(3) + w = 13 ⇒ 9 + w = 13 ⇒ w = 4.
• Final Answer: x = 1, y = 2, z = 3, w = 4

Exercise 3.2: Matrix Operations and Matrix Polynomials

Question 3: Matrix Multiplication and Polynomial Verification

Question: If A = [[3, 1], [−1, 2]], show that A² − 5A + 7I = O. Hence, find A⁻¹.

• Given: A = [[3, 1], [−1, 2]], I = [[1, 0], [0, 1]], O = [[0, 0], [0, 0]].
• To Prove: A² − 5A + 7I = O, and calculate A⁻¹.
• Step-by-Step Solution:
  1. Compute A² = A × A:
     A² = [[(3)(3) + (1)(−1), (3)(1) + (1)(2)], [(−1)(3) + (2)(−1), (−1)(1) + (2)(2)]]
     A² = [[9 − 1, 3 + 2], [−3 − 2, −1 + 4]] = [[8, 5], [−5, 3]]
  2. Compute −5A and 7I:
     5A = [[15, 5], [−5, 10]]
     7I = [[7, 0], [0, 7]]
  3. Substitute into A² − 5A + 7I:
     [[8, 5], [−5, 3]] − [[15, 5], [−5, 10]] + [[7, 0], [0, 7]]
     = [[8 − 15 + 7, 5 − 5 + 0], [−5 − (−5) + 0, 3 − 10 + 7]]
     = [[0, 0], [0, 0]] = O. (Hence Proved)
  4. Finding A⁻¹ using the matrix equation:
     A² − 5A + 7I = O
     Post-multiply throughout by A⁻¹ (since det(A) = 6 − (−1) = 7 ≠ 0):
     (A × A)A⁻¹ − 5AA⁻¹ + 7IA⁻¹ = O × A⁻¹
     A(AA⁻¹) − 5I + 7A⁻¹ = O
     AI − 5I + 7A⁻¹ = O ⇒ 7A⁻¹ = 5I − A
     7A⁻¹ = 5[[1, 0], [0, 1]] − [[3, 1], [−1, 2]] = [[5 − 3, 0 − 1], [0 − (−1), 5 − 2]] = [[2, −1], [1, 3]]
     A⁻¹ = (1/7) [[2, −1], [1, 3]]
• Final Answer: A⁻¹ = [[2/7, −1/7], [1/7, 3/7]]

Exercise 3.3: Transpose, Symmetric, and Skew-Symmetric Matrices

Question 4: Expressing a Matrix as the Sum of Symmetric and Skew-Symmetric Matrices

Question: Express the matrix B = [[2, −2, −4], [−1, 3, 4], [1, −2, −3]] as the sum of a symmetric and a skew-symmetric matrix.

• Formula / Theorem: Any square matrix B can be uniquely expressed as B = P + Q, where P = ½(B + B^T) is symmetric and Q = ½(B − B^T) is skew-symmetric.
• Step-by-Step Solution:
  1. Find the transpose B^T by swapping rows and columns:
     B^T = [[2, −1, 1], [−2, 3, −2], [−4, 4, −3]]
  2. Compute symmetric component P = ½(B + B^T):
     B + B^T = [[2+2, −2−1, −4+1], [−1−2, 3+3, 4−2], [1−4, −2+4, −3−3]] = [[4, −3, −3], [−3, 6, 2], [−3, 2, −6]]
     P = [[2, −3/2, −3/2], [−3/2, 3, 1], [−3/2, 1, −3]]
     Notice that P^T = P, so P is symmetric.
  3. Compute skew-symmetric component Q = ½(B − B^T):
     B − B^T = [[2−2, −2−(−1), −4−1], [−1−(−2), 3−3, 4−(−2)], [1−(−4), −2−4, −3−(−3)]] = [[0, −1, −5], [1, 0, 6], [5, −6, 0]]
     Q = [[0, −1/2, −5/2], [1/2, 0, 3], [5/2, −3, 0]]
     Notice that Q^T = −Q and diagonal elements are zero, so Q is skew-symmetric.
  4. Verification: P + Q = [[2+0, −3/2−1/2, −3/2−5/2], [−3/2+1/2, 3+0, 1+3], [−3/2+5/2, 1−3, −3+0]] = [[2, −2, −4], [−1, 3, 4], [1, −2, −3]] = B.
• Final Answer: B = [[2, −3/2, −3/2], [−3/2, 3, 1], [−3/2, 1, −3]] + [[0, −1/2, −5/2], [1/2, 0, 3], [5/2, −3, 0]]

Miscellaneous Exercise: Principle of Mathematical Induction (PMI) on Matrices

Question 5: Proof by Induction

Question: If A = [[cos θ, sin θ], [−sin θ, cos θ]], prove that Aⁿ = [[cos nθ, sin nθ], [−sin nθ, cos nθ]] for all n ∈ N.

• Given: A = [[cos θ, sin θ], [−sin θ, cos θ]].
• To Prove: P(n): Aⁿ = [[cos nθ, sin nθ], [−sin nθ, cos nθ]] ∀ n ∈ N.
• Step-by-Step Solution:
  1. Base Step (n = 1):
     P(1): A¹ = [[cos(1θ), sin(1θ)], [−sin(1θ), cos(1θ)]] = A. Thus, P(1) is true.
  2. Inductive Hypothesis:
     Assume P(k) is true for some positive integer k:
     A^k = [[cos kθ, sin kθ], [−sin kθ, cos kθ]].
  3. Inductive Step (To prove for n = k + 1):
     A^k+1 = A^k × A
     = [[cos kθ, sin kθ], [−sin kθ, cos kθ]] × [[cos θ, sin θ], [−sin θ, cos θ]]
     = [[cos kθ cos θ − sin kθ sin θ, cos kθ sin θ + sin kθ cos θ], [−sin kθ cos θ − cos kθ sin θ, −sin kθ sin θ + cos kθ cos θ]]
     Using trigonometric identities cos(A + B) = cos A cos B − sin A sin B and sin(A + B) = sin A cos B + cos A sin B:
     = [[cos(k + 1)θ, sin(k + 1)θ], [−sin(k + 1)θ, cos(k + 1)θ]] = P(k + 1).
  4. Conclusion: Since P(1) is true and P(k) ⇒ P(k + 1), by the Principle of Mathematical Induction, P(n) is true for all n ∈ N.
• Final Answer: Proved by Principle of Mathematical Induction for all n ∈ N.

Important Formulas and Theorems

Mastering NCERT Mathematics Class 12 solutions requires quick recall of core matrix algebra laws and definitions:

Common Mistakes and Tips

When solving CBSE NCERT solutions for Matrices, students frequently lose marks due to avoidable procedural slips:

• Assuming Matrix Multiplication is Commutative: Never write AB = BA or expand (A + B)² as A² + 2AB + B². The correct algebraic expansion is (A + B)² = A² + AB + BA + B².
• Forgetting the Zero Diagonal in Skew-Symmetric Matrices: For any skew-symmetric matrix M, m_ii = −m_ii ⇒ 2m_ii = 0 ⇒ m_ii = 0. If non-zero values appear on the principal diagonal, the matrix cannot be skew-symmetric.
• Order Compatibility in Multiplication: Before multiplying matrices A and B, always write down their orders. If A is 2 × 3 and B is 2 × 3, the product AB is undefined.
• Scalar Multiplication vs Determinant Scaling: In matrix scalar multiplication, k multiplies every single element of the matrix. (In determinants, k multiplies only a single row or column).
• Forgetting the Transpose Reversal Order: Always apply the reversal rule: (AB)^T = B^TA^T and (ABC)^T = C^TB^TA^T.

Board Exam Relevance and Marking Scheme

Matrices is one of the highest-scoring chapters in the CBSE Class 12 Mathematics curriculum. The questions from this chapter are direct, systematic, and follow predictable patterns:

• 1-Mark MCQs / Assertion-Reason: Order of product matrix, values of variables equating two matrices, skew-symmetric matrix diagonal properties, or calculation of k in equations like A² = kA − 2I.
• 2-Mark & 3-Mark Short Questions: Proving matrix equations (A² − 4A − 5I = O), finding A⁻¹ using matrix polynomials, or expressing a matrix as the sum of symmetric and skew-symmetric components.
• 4-Mark / 5-Mark Questions & Case Studies: Real-world cost matrices, production models, or induction-based proofs for Aⁿ.

More NCERT Solutions and Practice

To score full marks in your CBSE Class 12 Mathematics board exams, consistent practice with chapter-wise tests and real past board question formats is essential. Access verified chapter tests, step-by-step NCERT Solutions Mathematics chapter wise, and personalized AI-powered mock question papers at qptool.theorify.in.