NCERT Mathematics Class 12 - Chapter 3: Matrices - Notes

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: a_ij. Types: Row matrix (1 × n), Column matrix (m × 1), Square matrix (n × n), Diagonal matrix (a_ij = 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: (A_m×n)(B_n×p) = C_m×p; c_ij = Σ a_ikb_kj. 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

A^T (or A'): Interchange rows and columns. (A^T)^T = A. (A + B)^T = A^T + B^T. (kA)^T = kA^T. (AB)^T = B^TA^T (reverse order).

Special Matrices

Symmetric matrix: A^T = A (a_ij = a_ji). Skew-symmetric matrix: A^T = -A (a_ij = -a_ji; 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 + A^T)/2 + (A - A^T)/2.

Orthogonal matrix: AA^T = A^TA = I. Idempotent matrix: A² = A. Involutory matrix: A² = I. Nilpotent matrix: A^k = O for some positive integer k.

Elementary Operations and Invertible Matrices

Elementary row operations: (1) R_i ↔ R_j (interchange rows), (2) R_i → kR_i (multiply row by non-zero scalar), (3) R_i → R_i + kR_j (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 = B^TA^T
• Symmetric: A^T = A; Skew-symmetric: A^T = -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 = B^TA^T; (AB)^-1 = B^-1A^-1
• Symmetric: a_ij = a_ji; Skew-symmetric: a_ij = -a_ji, diagonal = 0
• A = (A+A^T)/2 + (A-A^T)/2 (sym + skew-sym decomposition)
• A invertible ⟺ |A| ≠ 0
• AB = O does NOT mean A = O or B = O