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Determinants

NCERT Class 12 Mathematics Chapter 4

Determinants for Class 12 Maths and JEE: determinants of order up to three, area of a triangle, minors, cofactors, adjoint, inverse and linear equations.

5 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 4 “Determinants”, reprint 2026-27.

Open the official NCERT chapter (PDF, 2.1 MB, ncert.nic.in)

Checked on 19 Oct 2026.

Learn in this order

  1. Determinant of a square matrix

    Determinant of a matrix for Class 12 Maths and JEE: 2 × 2 and 3 × 3, expanding along any row or column, |kA| = k|A|, the properties and solved examples.

    NCERT Sections 4.1–4.2 12 min read 5 solved examples 8 practice questions

  2. Area of a triangle using determinants

    Area of a triangle using determinants for Class 12 Maths and JEE: the formula, collinear points, the line through two points, with solved examples.

    NCERT Section 4.3 11 min read 4 solved examples 8 practice questions

  3. Minors and cofactors

    Minors and cofactors for Class 12 Maths and JEE: M and A = (−1) M, expansion by cofactors, the zero-sum rule and NCERT examples solved.

    NCERT Section 4.4 13 min read 5 solved examples 8 practice questions

  4. Adjoint and inverse of a matrix

    Adjoint and inverse of a matrix for Class 12 Maths and JEE: adj A, A(adj A) = |A| I, singular matrices, A = (1/|A|) adj A and NCERT examples solved.

    NCERT Section 4.5 14 min read 5 solved examples 12 practice questions

  5. Solving systems of linear equations with matrices

    System of linear equations for Class 12 Maths and JEE: matrix method X = AB, consistent and inconsistent systems, the |A| = 0 test and solved examples.

    NCERT Section 4.6 14 min read 5 solved examples 10 practice questions

Key formulas in this chapter

Determinant

  • |a b; c d| = ad − bc
  • order 3: expand along any row or column; each term is an element times its 2 × 2 determinant
  • sign of the (i, j) term: (−1)i+j
  • |kA| = k|A| for A of order n

Area of a triangle

  • area = ½ |x y 1; x y 1; x y 1|, taking the absolute value
  • collinear points ⇔ the determinant = 0
  • line through distinct (x, y), (x, y): |x y 1; x y 1; x y 1| = 0

Minors and cofactors

  • Aij = (−1)i+j Mij
  • Δ = a11A11 + a12A12 + a13A13 (any row or column works the same way)
  • a11A21 + a12A22 + a13A23 = 0 (a row with another row’s cofactors)

Adjoint and inverse

  • adj A = transpose of the matrix of cofactors Aij
  • A(adj A) = (adj A)A = |A| I
  • A = (1/|A|) adj A, for |A| ≠ 0
  • |AB| = |A||B| (same-order square matrices); |adj A| = |A| (n ≥ 2)

Solving linear systems

  • AX = B, with A the coefficient matrix
  • X = AB = (1/|A|)(adj A)B, for |A| ≠ 0
  • |A| = 0 and (adj A)B ≠ O: inconsistent

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