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NCERT Mathematics Class 12 - Chapter 3: Matrices - Notes

आव्यूह

Learning Objectives

  • Understand matrices, their types, and operations
  • Learn about transpose, symmetric, and skew-symmetric matrices
  • Study elementary row/column operations and invertible matrices

Key Concepts

Matrix Basics

A matrix is a rectangular array of numbers arranged in rows and columns. Order: m × n (m rows, n columns). Element in i-th row, j-th column: aij. Types: Row matrix (1 × n), Column matrix (m × 1), Square matrix (n × n), Diagonal matrix (aij = 0 for i ≠ j), Scalar matrix (diagonal with all diagonal elements equal), Identity matrix I (diagonal with all 1s), Zero/Null matrix (all elements 0).

Operations on Matrices

Addition: A + B (same order; add corresponding elements). Scalar multiplication: kA (multiply each element by k). Matrix multiplication: (Am×n)(Bn×p) = Cm×p; cij = Σ aikbkj. Matrix multiplication is not commutative (AB ≠ BA in general) but is associative (A(BC) = (AB)C) and distributive (A(B+C) = AB + AC).

Important: AB = O does not imply A = O or B = O. AB = AC does not imply B = C.

Transpose

AT (or A'): Interchange rows and columns. (AT)T = A. (A + B)T = AT + BT. (kA)T = kAT. (AB)T = BTAT (reverse order).

Special Matrices

Symmetric matrix: AT = A (aij = aji). Skew-symmetric matrix: AT = -A (aij = -aji; diagonal elements must be 0). Decomposition theorem: Every square matrix A can be uniquely written as sum of symmetric and skew-symmetric matrices: A = (A + AT)/2 + (A - AT)/2.

Orthogonal matrix: AAT = ATA = I. Idempotent matrix: A2 = A. Involutory matrix: A2 = I. Nilpotent matrix: Ak = O for some positive integer k.

Elementary Operations and Invertible Matrices

Elementary row operations: (1) Ri ↔ Rj (interchange rows), (2) Ri → kRi (multiply row by non-zero scalar), (3) Ri → Ri + kRj (add multiple of one row to another). A square matrix A is invertible if there exists A-1 such that AA-1 = A-1A = I. A is invertible iff |A| ≠ 0. (AB)-1 = B-1A-1.

Summary

Matrices are rectangular arrays with defined operations. Multiplication is non-commutative but associative. Every square matrix decomposes into symmetric and skew-symmetric parts. Invertible matrices satisfy AA-1 = I and require non-zero determinant.

Important Terms

  • Identity matrix: Diagonal matrix with all 1s; AI = IA = A
  • Transpose: Rows become columns; (AB)T = BTAT
  • Symmetric: AT = A; Skew-symmetric: AT = -A
  • Invertible: A-1 exists iff |A| ≠ 0
  • Idempotent: A2 = A

Quick Revision

  • Matrix multiplication: columns of A must equal rows of B
  • AB ≠ BA (not commutative); (AB)C = A(BC) (associative)
  • (AB)T = BTAT; (AB)-1 = B-1A-1
  • Symmetric: aij = aji; Skew-symmetric: aij = -aji, diagonal = 0
  • A = (A+AT)/2 + (A-AT)/2 (sym + skew-sym decomposition)
  • A invertible ⟺ |A| ≠ 0
  • AB = O does NOT mean A = O or B = O
NCERT Mathematics Class 12 - Chapter 3: Matrices - Notes | EduMunch