Matrices & Determinants

Concept overview

A matrix is a rectangular array of numbers. Addition, scalar multiplication and matrix multiplication have standard rules; AB is defined when columns of A match rows of B, and generally AB ≠ BA. Special matrices (identity I, zero, diagonal, symmetric, skew-symmetric, orthogonal) each have distinctive properties.

The determinant of a square matrix is a scalar with important geometric meaning: |A| is the signed volume of the parallelepiped spanned by rows. Key properties: det(AB) = det(A) det(B), det(A^T) = det(A), swapping two rows flips the sign, scaling a row by k scales det by k.

A square matrix A is invertible iff det(A) ≠ 0. The inverse formula A⁻¹ = adj(A) / det(A) uses the adjoint (transpose of cofactor matrix). Cramer's rule solves a linear system Ax = b directly through determinants: xᵢ = det(Aᵢ) / det(A), where Aᵢ replaces column i of A with b.

Rank of a matrix is the number of linearly independent rows (or columns — they're equal). For a system Ax = b: unique solution iff rank(A) = rank(A|b) = n; no solution iff rank(A) < rank(A|b); infinitely many iff rank(A) = rank(A|b) < n.

Key formulas & concepts

• det(AB): = det(A) × det(B)
• det(A^T): = det(A)
• Inverse formula: A⁻¹ = adj(A) / det(A)
• Cramer's rule: xᵢ = det(Aᵢ) / det(A)
• Symmetric matrix: A^T = A
• Skew-symmetric: A^T = −A; diagonal entries are 0
• Orthogonal matrix: A^T A = I; det(A) = ±1

Common mistakes to avoid

• Assuming AB = BA — matrix multiplication is non-commutative.
• Using Cramer's rule on a system with det(A) = 0 — no unique solution.
• Computing det of a non-square matrix — determinants are defined only for square matrices.
• Confusing rank with dimension — dimension is a property of a vector space; rank is a property of a matrix.

JEE Main relevance

JEE Main asks 2 matrices/determinant problems every paper — usually one determinant property or inverse computation, and one Cramer's rule or system-solvability question.

JEE Advanced relevance

Advanced problems combine matrix identities (e.g., (I + A)² for a specific A), or ask about consistency of systems with parameters, or use eigenvalue-like reasoning without explicitly naming eigenvalues.

Previous-year questions (PYQ)

Determinant properties and inverse computations dominate the PYQ pool. Drill both in a chapter-wise session.