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Matrices

NCERT Class 12 Mathematics Chapter 3

Matrices for Class 12 Maths and JEE: order and types of matrices, addition, scalar and matrix multiplication, transpose, symmetric matrices and inverses.

6 topics Updated 8 Nov 2026

For: Class 12 and JEE

In the syllabus: JEE Class 12 syllabus

With your textbook

NCERT Mathematics Part I (Class 12), Chapter 3 “Matrices”, reprint 2026-27.

Open the official NCERT chapter (PDF, 3.4 MB, ncert.nic.in)

Checked on 19 Oct 2026.

Learn in this order

  1. Matrices and their order

    What a matrix is, for Class 12 Maths and JEE: rows, columns, the element a, the order m × n, possible orders, and building a matrix from a rule.

    NCERT Sections 3.1–3.2 12 min read 5 solved examples 9 practice questions

  2. Types of matrices

    Types of matrices for Class 12 Maths and JEE: row, column, square, diagonal, scalar, identity and zero matrices, and when two matrices are equal.

    NCERT Section 3.3 12 min read 4 solved examples 8 practice questions

  3. Operations on matrices

    Operations on matrices for Class 12 Maths and JEE: addition, scalar multiples, subtraction, matrix multiplication, its properties and why AB ≠ BA.

    NCERT Section 3.4 17 min read 6 solved examples 17 practice questions

  4. Transpose of a matrix

    Transpose of a matrix for Class 12 Maths and JEE: how rows become columns, its properties, why (AB)′ = B′A′, and solved 2 × 2 and 3 × 3 examples.

    NCERT Section 3.5 11 min read 5 solved examples 9 practice questions

  5. Symmetric and skew symmetric matrices

    Symmetric and skew symmetric matrices for Class 12 Maths and JEE: definitions, properties, the zero diagonal, any square matrix as their sum, examples.

    NCERT Section 3.6 10 min read 5 solved examples 10 practice questions

  6. Invertible matrices

    Invertible matrices for Class 12 Maths and JEE: what the inverse of a matrix is, why it is unique, (AB) = BA, how to find it, with solved examples.

    NCERT Section 3.7 13 min read 4 solved examples 9 practice questions

Key formulas in this chapter

Matrix and order

  • A = [aij] of order m × n: m rows, n columns
  • aij = element in row i and column j
  • number of elements = mn

Types of matrices

  • column matrix: m × 1; row matrix: 1 × n; square matrix: m = n
  • diagonal: aij = 0 for i ≠ j
  • scalar: diagonal with aii = k; identity I: k = 1
  • A = B ⇔ same order and aij = bij for all i, j

Operations on matrices

  • A + B = [aij + bij]; kA = [kaij]; A − B = A + (−1)B
  • AB: (m × n)(n × p) gives m × p
  • cik = ai1b1k + ai2b2k + … + ainbnk
  • (AB)C = A(BC); A(B + C) = AB + AC; I A = A I = A
  • in general AB ≠ BA

Transpose of a matrix

  • A = [aij] of order m × n ⇒ A′ = [aji] of order n × m
  • (A′)′ = A; (kA)′ = kA′
  • (A + B)′ = A′ + B′
  • (AB)′ = B′A′

Symmetric and skew symmetric

  • symmetric: A′ = A; skew symmetric: A′ = −A, so aii = 0
  • A + A′ is symmetric; A − A′ is skew symmetric
  • A = ½(A + A′) + ½(A − A′)

Invertible matrices

  • AA = AA = I
  • (AB) = BA
  • (A) = A

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