Board exam focus — Matrices (CBSE & HBSE)

Matrices carries 8-10 marks in CBSE and HBSE. Questions span operations (addition, multiplication), transpose, symmetric/skew-symmetric decomposition, elementary operations, and finding inverse using elementary row operations.

Types of Matrices and Matrix Operations

Types of Matrices

Matrix Addition

For A and B of same order m×n: (A+B)ᵢⱼ = aᵢⱼ + bᵢⱼ

Properties:

• A+B = B+A (commutative)
• (A+B)+C = A+(B+C) (associative)
• A+O = O+A = A (O is additive identity)
• A+(−A) = O (additive inverse)

Scalar Multiplication

(kA)ᵢⱼ = k·aᵢⱼ

Matrix Multiplication

For A of order m×n and B of order n×p, AB exists and is m×p.

(AB)ᵢⱼ = Σₖ aᵢₖ · bₖⱼ (sum over k from 1 to n)

Properties of Matrix Multiplication:

• In general, AB ≠ BA (NOT commutative)
• (AB)C = A(BC) (associative)
• A(B+C) = AB+AC (distributive)
• AI = IA = A (I is multiplicative identity)
• AB = 0 does NOT necessarily mean A = 0 or B = 0

Order compatibility rule: For AB to exist, columns of A = rows of B. If A is m×n and B is n×p, then AB is m×p.

CBSE Tip: Always check the order before multiplying. The product AB has the same number of rows as A and same number of columns as B.

Transpose, Symmetric and Skew-Symmetric Matrices

Transpose of a Matrix

The transpose A' (or Aᵀ) of an m×n matrix A is obtained by interchanging rows and columns. (A')ᵢⱼ = Aⱼᵢ

Properties of Transpose:

• (A')' = A
• (kA)' = kA'
• (A+B)' = A'+B'
• (AB)' = B'A' (order reverses!)

Symmetric Matrix

A square matrix A is symmetric iff A' = A ⟺ aᵢⱼ = aⱼᵢ for all i, j

Example: A = [[1,2,3],[2,4,5],[3,5,6]] is symmetric.

Skew-Symmetric Matrix

A square matrix A is skew-symmetric iff A' = −A ⟺ aᵢⱼ = −aⱼᵢ for all i, j

Key result: The diagonal entries of a skew-symmetric matrix are all zero (since aᵢᵢ = −aᵢᵢ ⟹ 2aᵢᵢ = 0).

Decomposition Theorem

Every square matrix A can be written as: A = (1/2)(A+A') + (1/2)(A−A')

where (1/2)(A+A') is symmetric and (1/2)(A−A') is skew-symmetric.

Results on Symmetric Matrices

• If A is symmetric, A+A' = 2A (symmetric), A−A' = O
• If A and B are symmetric: A+B is symmetric; AB is symmetric iff AB = BA
• AA' and A'A are always symmetric

CBSE Tip: To show a matrix is symmetric, compute A' and verify A' = A. To decompose a matrix, always use the formula A = P + Q where P = (A+A')/2 and Q = (A−A')/2.

Elementary Row Operations and Finding Inverse

Elementary Row Operations

Three types of elementary row operations (ERO) on a matrix:

1. Rᵢ ↔ Rⱼ — interchange rows i and j
2. Rᵢ → kRᵢ — multiply row i by non-zero scalar k
3. Rᵢ → Rᵢ + kRⱼ — add k times row j to row i

Finding A⁻¹ by Elementary Row Operations

Method: Write [A | I] and apply EROs to convert A to I. The same EROs convert I to A⁻¹.

Existence Condition: A⁻¹ exists iff |A| ≠ 0 (A is non-singular).

Step-by-step process:

1. Write the augmented matrix [A | I]
2. Apply EROs to reduce A to I
3. The right part becomes A⁻¹

Example: Find A⁻¹ for A = [[2,1],[5,3]]

• [A|I] = [[2,1,1,0],[5,3,0,1]]
• R₂ → R₂ − (5/2)R₁: [[2,1,1,0],[0,1/2,−5/2,1]]
• R₂ → 2R₂: [[2,1,1,0],[0,1,−5,2]]
• R₁ → R₁ − R₂: [[2,0,6,−2],[0,1,−5,2]]
• R₁ → R₁/2: [[1,0,3,−1],[0,1,−5,2]]
• ∴ A⁻¹ = [[3,−1],[−5,2]]

Verify: AA⁻¹ = [[2·3+1·(−5), 2·(−1)+1·2],[5·3+3·(−5), 5·(−1)+3·2]] = [[1,0],[0,1]] ✓

Properties of Invertible Matrices

• (A⁻¹)⁻¹ = A
• (AB)⁻¹ = B⁻¹A⁻¹
• (A')⁻¹ = (A⁻¹)'
• (kA)⁻¹ = (1/k)A⁻¹

CBSE Tip: For 2×2 matrix [[a,b],[c,d]], A⁻¹ = (1/|A|)[[d,−b],[−c,a]]. This is a quick formula. For 3×3, use ERO or adjoint method.

Frequently asked questions

Are these Matrices notes free?

Yes — the Matrices notes for Mathematics (Class 12) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Matrices notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Matrices chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Matrices.