Chapter 3 — Matrices — covers types of matrices, operations, transpose, symmetric/skew-symmetric matrices, and elementary transformations. Carries 5-7 marks.
Key Concepts
Types of Matrices
| Type | Definition |
|---|---|
| Row Matrix | Only 1 row (1 × n) |
| Column Matrix | Only 1 column (m × 1) |
| Square Matrix | Number of rows = number of columns (n × n) |
| Diagonal Matrix | All non-diagonal elements are 0 |
| Identity Matrix (I) | Diagonal matrix with all diagonal elements = 1 |
| Zero Matrix (O) | All elements are 0 |
| Symmetric | A = Aᵀ (aᵢⱼ = aⱼᵢ) |
| Skew-Symmetric | A = −Aᵀ (aᵢⱼ = −aⱼᵢ, diagonal = 0) |
Matrix Operations
Addition: A + B (same order, add corresponding elements)
Scalar multiplication: kA (multiply every element by k)
Matrix multiplication: (A)ₘₓₙ × (B)ₙₓₚ = (C)ₘₓₚ
cᵢⱼ = Σ aᵢₖ × bₖⱼ (row of A × column of B)
Important: AB ≠ BA in general (not commutative!)
But: (AB)C = A(BC) — associative ✓
A(B+C) = AB + AC — distributive ✓
Scalar multiplication: kA (multiply every element by k)
Matrix multiplication: (A)ₘₓₙ × (B)ₙₓₚ = (C)ₘₓₚ
cᵢⱼ = Σ aᵢₖ × bₖⱼ (row of A × column of B)
Important: AB ≠ BA in general (not commutative!)
But: (AB)C = A(BC) — associative ✓
A(B+C) = AB + AC — distributive ✓
Transpose Properties
(Aᵀ)ᵀ = A
(A + B)ᵀ = Aᵀ + Bᵀ
(kA)ᵀ = kAᵀ
(AB)ᵀ = BᵀAᵀ (order reverses!)
(A + B)ᵀ = Aᵀ + Bᵀ
(kA)ᵀ = kAᵀ
(AB)ᵀ = BᵀAᵀ (order reverses!)
Key Result: Every square matrix can be uniquely expressed as sum of a symmetric and skew-symmetric matrix:
A = ½(A + Aᵀ) + ½(A − Aᵀ)
A = ½(A + Aᵀ) + ½(A − Aᵀ)
Solved Examples
Example 1
Q: If A is a 3×2 matrix and B is a 2×4 matrix, what is the order of AB?
Solution: (3×2)(2×4) = 3×4 matrix. The inner dimensions (2) must match; result has outer dimensions.
Example 2
Q: Express the matrix A = [1 2; 3 4] as sum of symmetric and skew-symmetric matrices.
Solution:
Aᵀ = [1 3; 2 4]
Symmetric part = ½(A + Aᵀ) = ½[2 5; 5 8] = [1 5/2; 5/2 4]
Skew-symmetric part = ½(A − Aᵀ) = ½[0 −1; 1 0] = [0 −1/2; 1/2 0]
Quick Revision Points
- Matrix multiplication: order m×n × n×p = m×p; inner dimensions must match
- AB ≠ BA (not commutative); (AB)ᵀ = BᵀAᵀ
- Symmetric: A = Aᵀ; Skew-symmetric: A = −Aᵀ (diagonal = 0)
- A = ½(A+Aᵀ) + ½(A−Aᵀ) — decomposition into symmetric + skew-symmetric
- AI = IA = A (identity matrix property)
Chapter Navigation
Previous: Inverse Trigonometric Functions Class 12 Notes
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Related Chapters in Class 12 Maths
- Determinants Class 12 Notes
- Relations and Functions Class 12 Notes
- Continuity and Differentiability Class 12 Notes
Practice What You Learned
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