Matrices
Learn Matrices with notes, examples, and practice questions.
Chapter Overview
A matrix is a rectangular array of numbers arranged in rows and columns. Matrices compactly represent linear transformations, systems of equations, and data.
Class 12 covers matrix algebra: addition, multiplication, transpose, and special matrix types.
Matrix Operations
Addition
Same order required — add corresponding entries
Scalar multiply
kA: multiply every entry by k
Multiplication
AB defined if cols of A = rows of B; (AB)ᵢⱼ = Σₖ aᵢₖbₖⱼ
- AB ≠ BA in general (not commutative)
- A(BC) = (AB)C (associative)
- A(B+C) = AB + AC (distributive)
- Iₙ is identity: AI = IA = A
Transpose and Special Matrices
- Transpose (A′): rows ↔ columns
- (AB)′ = B′A′
- Symmetric: A′ = A
- Skew-symmetric: A′ = −A → diagonal entries are 0
- Zero matrix, scalar matrix, diagonal matrix
Elementary Row/Column Operations
- Rᵢ ↔ Rⱼ or Cᵢ ↔ Cⱼ (interchange)
- Rᵢ → kRᵢ or Cᵢ → kCᵢ (scaling)
- Rᵢ → Rᵢ + kRⱼ (add multiple of one row to another)
- Used to reduce matrices to echelon form — foundation for determinants and inverses
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Matrix Multiplication
If A = [[1,2],[0,3]] and B = [[4,1],[2,0]], find AB.
- 1Row 1: [1·4+2·2, 1·1+2·0] = [8, 1]
- 2Row 2: [0·4+3·2, 0·1+3·0] = [6, 0]
Answer
AB = [[8,1],[6,0]]
Example 2: Transpose
If A = [[1,−2],[3,4]], find A′ and check if symmetric.
- 1A′ = [[1,3],[−2,4]]
- 2A ≠ A′ → not symmetric
Answer
A′ = [[1,3],[−2,4]]; not symmetric
Example 3: Skew-Symmetric
Show A = [[0,5],[−5,0]] is skew-symmetric.
- 1A′ = [[0,−5],[5,0]] = −A
- 2Diagonal zeros required for skew-symmetric
Answer
A′ = −A ✓
Key Points to Remember
- ✓ Order of A(m×n) × B(n×p) = C(m×p)
- ✓ Only square matrices can have inverses (next chapter)
- ✓ Skew-symmetric matrices always have zero diagonal
Exam Tips
- • Check orders before multiplying
- • Show each row-column dot product in board exams
- • Symmetric + skew-symmetric decomposition appears in proofs
Formula Cheat Sheet
Quick reference — NCERT board exam formulas
Matrix Algebra
- (A ± B)ᵀ = Aᵀ ± Bᵀ
- (kA)ᵀ = kAᵀ ; (AB)ᵀ = BᵀAᵀ
- (Aᵀ)ᵀ = A
- (A⁻¹)ᵀ = (Aᵀ)⁻¹
- A⁻¹ = (1/|A|) adj(A), |A| ≠ 0
- |AB| = |A||B| ; |Aᵀ| = |A|
- adj(AB) = adj(B) adj(A)
Practice MCQs — Matrices
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
If A is 2×3 and B is 3×4, order of AB is: