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Schooling · Class 12 · Mathematics · Determinants · Chapter on a page

CBSE Class 12 · Mathematics · NCERT Mathematics Part-I

Determinants: chapter on a page

A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).

About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.

Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain.

Mind map

Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.

Mind map of Determinants: Key ideas, Formulas and facts, Watch out forKey ideasThe determinant of a 2 × 2 matrix A…For a 3 × 3 matrix, the determinant…The minor M_ij of an element a_ij is…Key propertiesA square matrix is singular…The adjoint of a matrix A, written…Cramer's rule gives the solution of…Formulas and factsdet([[a, b], [c, d]]) = ad − bc…For a 3 × 3 matrix [[a₁, b₁, c₁]…|kA| = k^n × |A| for an n × n matrix A|AB| = |A| × |B| for square matrices…|A^T| = |A| — the determinant of the…adj(A) × A = A × adj(A) = |A| × I…A⁻¹ = (1/|A|) × adj(A)Watch out forForgetting the sign pattern in…Multiplying all elements by k and…Adding a multiple of one row to…Confusing the adjoint with the inverseUsing Cramer's rule when the…Determinants
The mind map as a list
  • Key ideas
    • The determinant of a 2 × 2 matrix A…
    • For a 3 × 3 matrix, the determinant…
    • The minor M_ij of an element a_ij is…
    • Key properties
    • A square matrix is singular…
    • The adjoint of a matrix A, written…
    • Cramer's rule gives the solution of…
  • Formulas and facts
    • det([[a, b], [c, d]]) = ad − bc…
    • For a 3 × 3 matrix [[a₁, b₁, c₁]…
    • |kA| = k^n × |A| for an n × n matrix A
    • |AB| = |A| × |B| for square matrices…
    • |A^T| = |A| — the determinant of the…
    • adj(A) × A = A × adj(A) = |A| × I…
    • A⁻¹ = (1/|A|) × adj(A)
  • Watch out for
    • Forgetting the sign pattern in…
    • Multiplying all elements by k and…
    • Adding a multiple of one row to…
    • Confusing the adjoint with the inverse
    • Using Cramer's rule when the…

What this chapter is about

Determinants are special numerical values associated with square matrices. After learning matrices in the previous chapter, you now study determinants because they reveal important properties of a matrix — most crucially, whether a system of linear equations has a unique solution. A determinant converts an entire square array of numbers into a single number that carries geometric and algebraic meaning.

Key ideas

  • The determinant of a 2 × 2 matrix A = [[a, b], [c, d]] is written |A| or det(A) and equals ad − bc.
  • For a 3 × 3 matrix, the determinant is computed by expansion along any row or column, using minors and cofactors.
  • The minor M_ij of an element a_ij is the determinant of the 2 × 2 matrix obtained by deleting the i-th row and j-th column. The cofactor is C_ij = (−1)^(i+j) × M_ij.
  • Key properties: interchanging two rows (or columns) changes the sign; if two rows are identical, the determinant is zero; multiplying a row by k multiplies the determinant by k; adding a multiple of one row to another leaves the determinant unchanged.
  • A square matrix is singular (non-invertible) if and only if its determinant is zero.
  • The adjoint of a matrix A, written adj(A), is the transpose of the matrix of cofactors. The inverse is A⁻¹ = (1/|A|) × adj(A), valid when |A| ≠ 0.
  • Cramer's rule gives the solution of a system of n linear equations in n unknowns using ratios of determinants, provided the coefficient determinant is non-zero.
  • The area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) equals (1/2)|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.

Formulas and facts to remember

  • det([[a, b], [c, d]]) = ad − bc — the product of the main diagonal minus the product of the other diagonal.
  • For a 3 × 3 matrix [[a₁, b₁, c₁], [a₂, b₂, c₂], [a₃, b₃, c₃]], expanding along the first row: |A| = a₁(b₂c₃ − b₃c₂) − b₁(a₂c₃ − a₃c₂) + c₁(a₂b₃ − a₃b₂).
  • |kA| = k^n × |A| for an n × n matrix A.
  • |AB| = |A| × |B| for square matrices of the same order.
  • |A^T| = |A| — the determinant of the transpose equals the determinant itself.
  • adj(A) × A = A × adj(A) = |A| × I, where I is the identity matrix.
  • A⁻¹ = (1/|A|) × adj(A), provided |A| ≠ 0.
  • Cramer's rule for two variables: if ax + by = e and cx + dy = f, then x = (ed − bf)/(ad − bc), y = (af − ec)/(ad − bc), when ad − bc ≠ 0.

Common mistakes

  • Forgetting the sign pattern in cofactors → remember C_ij = (−1)^(i+j) × M_ij; the checkerboard starts with + at position (1,1).
  • Multiplying all elements by k and thinking |kA| = k × |A| → for an n × n matrix, |kA| = k^n × |A|.
  • Adding a multiple of one row to another and also multiplying the determinant by that factor → this operation leaves the determinant unchanged; only multiplying a single row by k changes the value.
  • Confusing the adjoint with the inverse → adj(A) is the transpose of the cofactor matrix; the inverse additionally divides by |A|.
  • Using Cramer's rule when the coefficient determinant is zero → the rule fails; the system has either no solution or infinitely many.

Quick revision

  • Determinant of a 2 × 2: ad − bc; of a 3 × 3: expand using cofactors along any row or column.
  • Swapping two rows changes the sign; identical rows give zero determinant.
  • A matrix is invertible exactly when its determinant is non-zero.
  • Inverse formula: A⁻¹ = (1/|A|) × adj(A).
  • Area of triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) uses a determinant formula and equals half the absolute value.