AlgebraTop Priority (3-4 Qs in JEE)

Matrices & Determinants Formulas

Matrix algebra, determinant expansion, adjoint properties, inverse, Cramer's rule, and systems of linear equations.

9 Formulas•1,015+ Questions in Bank

Core Formulas & Mathematical Identities

|A^T| = |A|, \quad |AB| = |A||B|, \quad |kA| = k^n |A| \quad (n \times n \text{ matrix})

Scalar multiplication pulls out factor k raised to the dimension power n.

Examiner Tip: |A⁻¹| = 1 / |A|.

A \cdot \text{adj}(A) = |A| I_n, \quad |\text{adj}(A)| = |A|^{n-1}, \quad |\text{adj}(\text{adj}(A))| = |A|^{(n-1)^2}

Crucial adjoint determinant power formulas tested almost every year in JEE Main.

Examiner Tip: For 3x3 matrix (n = 3): |adj(A)| = |A|² and |adj(adj(A))| = |A|⁴.

A^{-1} = \frac{1}{|A|} \text{adj}(A) \quad (|A| \ne 0)

Inverse exists if and only if matrix A is non-singular (|A| ≠ 0).

Examiner Tip: (AB)⁻¹ = B⁻¹ A⁻¹ (reversal law).

x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}, \quad z = \frac{\Delta_z}{\Delta}

Determinant method for solving system of linear equations.

Examiner Tip: Δ ≠ 0: Unique solution. Δ = 0 with any Δ_i ≠ 0: Inconsistent (No solution).

A = \frac{A + A^T}{2} + \frac{A - A^T}{2}, \quad \text{Skew-Symmetric: } A^T = -A \implies a_{ii} = 0

Every square matrix is uniquely decomposable into symmetric and skew-symmetric parts.

Examiner Tip: Determinant of odd-order skew-symmetric matrix is ALWAYS zero.

A \cdot A^T = I \implies |A| = \pm 1, \quad A^{-1} = A^T

Orthogonal matrices preserve lengths and angles.

Examiner Tip: Rows and columns form orthonormal vectors.

\det(A - \lambda I) = 0 \implies P(A) = O, \quad A^2 - \text{tr}(A)A + |A|I = O \; (2 \times 2)

Every square matrix satisfies its own characteristic polynomial equation. Enables computing high powers and inverses.

Examiner Tip: To find A⁻¹, multiply the characteristic equation throughout by A⁻¹.

A^2 = A \; (\text{Idempotent}), \quad A^2 = I \; (\text{Involutory}), \quad A^k = O \; (\text{Nilpotent}), \quad \text{tr}(AB) = \text{tr}(BA)

Key algebraic matrix classes frequently tested in eigenvalue and matrix polynomial questions.

Examiner Tip: Eigenvalues of idempotent matrices can only be 0 or 1. Determinant of involutory matrix is ±1.

|A| \ne 0 \implies X = O \; (\text{Trivial Unique Solution}), \quad |A| = 0 \implies \text{Infinitely Many Non-Trivial Solutions}

Criterion for existence of non-zero solutions to homogeneous linear equations.

Examiner Tip: If a homogeneous system has a non-trivial solution, its determinant MUST equal 0.

Frequently tested misconceptions in Matrices & Determinants

•Confusing det(kA) with k*det(A): det(kA) = k^n * det(A) where n is matrix order.
•Cramer's rule: if Δ = 0 and at least one Δ_i ≠ 0, NO solution. If all Δ = Δ_x = Δ_y = Δ_z = 0, can be infinite OR no solution.