CBSE Class 12 Maths: Matrices & Determinants — Notes 2026

Matrix Operations

• Addition: same order matrices; commutative and associative; zero matrix is additive identity
• Multiplication: AB requires columns of A = rows of B; not commutative; associative
• Transpose properties: (A+B)ᵀ = Aᵀ+Bᵀ; (AB)ᵀ = BᵀAᵀ; (kA)ᵀ = kAᵀ

Special Matrices

• Symmetric: A = Aᵀ; skew-symmetric: A = −Aᵀ; diagonal elements of skew-symmetric are 0
• Every square matrix = ½(A+Aᵀ) + ½(A−Aᵀ) = symmetric + skew-symmetric
• Idempotent: A² = A; involutory: A² = I; nilpotent: Aⁿ = O for some positive integer n

Determinants of 2×2 and 3×3

• |A| = ad−bc for 2×2; expansion along any row or column for 3×3 using cofactors
• Properties: |AB| = |A||B|; |Aᵀ| = |A|; if two rows/columns equal, |A| = 0; scalar factor: |kA| = kⁿ|A|
• Singular matrix: |A| = 0; non-singular: |A| ≠ 0; non-singular matrix has an inverse

Adjoint and Inverse of a Matrix

• Adjoint (adj A): transpose of cofactor matrix; A × adj(A) = adj(A) × A = |A| × I
• Inverse: A⁻¹ = adj(A)/|A|; exists only when |A| ≠ 0 (non-singular)
• Properties: (AB)⁻¹ = B⁻¹A⁻¹; (Aᵀ)⁻¹ = (A⁻¹)ᵀ; (A⁻¹)⁻¹ = A

System of Linear Equations

• Matrix form: AX = B; if |A| ≠ 0, unique solution: X = A⁻¹B (non-singular)
• Cramer's rule: x = D_x/D, y = D_y/D, z = D_z/D where D = |A|, D_x = |replace col-1 with B|
• If D = 0 and D_x = D_y = D_z = 0: infinite solutions; if D = 0 and any D_x ≠ 0: no solution

Area of Triangle Using Determinants

• Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| = ½|det of 3×3 matrix with third column = 1|
• Collinear condition: area = 0 → determinant = 0
• CBSE application: find value of k for collinearity; find unknown vertex of triangle with given area

CBSE Board Focus

• Matrices and Determinants: 8–10 marks (most important chapter in Class 12 Maths)
• Inverse using adjoint method: write all steps; verify A.A⁻¹ = I; CBSE gives partial marks for steps
• Solve system of 3 equations using matrix method: full 6-mark question frequently asked