Matrices and Determinants

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Rapid summary for last-minute revision before your exam.

Matrices and Determinants — Key Facts for JEE Main Matrix: rectangular array of numbers in rows and columns Square matrix: same number of rows and columns (n × n) Determinant: scalar value computed from square matrix For 2×2 matrix A = [[a, b], [c, d]], det(A) = ad − bc Singular matrix: det(A) = 0; Non-singular: det(A) ≠ 0 Inverse: A⁻¹ = adj(A)/det(A) (only for non-singular matrices) ⚡ Exam tip: JEE Main tests matrix operations and determinant properties frequently — especially adjoint, inverse, and solving linear equations!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Matrices and Determinants — JEE Main Study Guide

Matrix operations:

• Addition: A + B works only when dimensions match
• Scalar multiplication: k·A multiplies each element by k
• Multiplication: A_{m×n} × B_{n×p} = C_{m×p}; number of columns in A must equal rows in B
• Transpose: A^T switches rows and columns

Determinant properties:

• det(A^T) = det(A)
• det(AB) = det(A)·det(B)
• det(kA) = k^n · det(A) for n×n matrix
• Swapping two rows changes sign of determinant
• Adding multiple of one row to another leaves det unchanged
• If two rows are identical, det = 0

Adjoint: adj(A) = Cofactor matrix transpose For 3×3: adj(A) is the transpose of the matrix of cofactors

Inverse via adjoint: A⁻¹ = adj(A)/det(A) — verify that A·A⁻¹ = I

Solving linear equations using matrices: AX = B → X = A⁻¹B (when A is non-singular) Cramer’s rule: for 2 equations, x = D_x/D, y = D_y/D where D is determinant of coefficients

Types of matrices:

• Symmetric: A^T = A; a_ij = a_ji
• Skew-symmetric: A^T = −A; a_ii = 0
• Orthogonal: A·A^T = I; det = ±1
• Idempotent: A² = A
• Nilpotent: A^k = 0 for some k

Rank of matrix:

• Row rank = column rank

• Non-zero rows after row reduction give rank

• det(A) ≠ 0 → full rank = n for n×n matrix

• Key formula: det(A) = ad − bc for 2×2; A⁻¹ = adj(A)/det(A)

• Common trap: AB = 0 does not imply A = 0 or B = 0 (zero divisors exist in matrices)

• Exam weight: 1 question per year (4 marks); often combined with solving linear equations or Eigenvalues

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Matrices and Determinants — Comprehensive JEE Main Notes

Determinant expansion (Laplacian): For 3×3 matrix, expand along first row: det(A) = a₁·C₁₁ − b₁·C₁₂ + c₁·C₁₃ Where C_ij is the cofactor = (−1)^{i+j} · M_ij (M_ij = minor, determinant of matrix after removing i-th row and j-th column)

Eigenvalues and Eigenvectors: For matrix A, λ is eigenvalue if det(A − λI) = 0 Characteristic equation: |A − λI| = 0 For 2×2: λ² − (trace)λ + det = 0 Eigenvector v satisfies: A·v = λ·v

Properties:

• Sum of eigenvalues = trace(A) = a₁₁ + a₂₂ + …
• Product of eigenvalues = det(A)
• Similar matrices (A and PAP⁻¹) have same eigenvalues

System of linear equations: For n equations in n variables:

• Unique solution: rank(A) = rank([A|B]) = n
• Infinite solutions: rank(A) = rank([A|B]) < n
• No solution: rank(A) ≠ rank([A|B])

Using row reduction for determinants: Use elementary row operations (which don’t change determinant or multiply by factor) to simplify If you multiply a row by k, determinant multiplies by k (not preserve) If you add multiple of one row to another, determinant unchanged ✓

Matrix equations: Solve AX = B: multiply both sides by A⁻¹ (if it exists): X = A⁻¹B Solve XA = B: multiply both sides by A⁻¹: X = BA⁻¹ Note: AX = B and XA = B give different results in general!

Special matrices:

• For symmetric matrix A, A^T = A
• For skew-symmetric A, A^T = −A and diagonal elements are 0
• Every square matrix can be expressed as sum of symmetric and skew-symmetric parts: A = (A + A^T)/2 + (A − A^T)/2
• Hermitian: A^dagger = A (conjugate transpose)

Properties of orthogonal matrices: A·A^T = I; therefore A⁻¹ = A^T det(A) = ±1 If det = +1: proper rotation If det = −1: improper rotation/reflection

Inverse property for 2×2: For A = [[a, b], [c, d]], A⁻¹ = (1/(ad−bc))·[[d, −b], [−c, a]]

Cayley-Hamilton theorem: Every matrix satisfies its own characteristic equation: For 2×2 with characteristic equation λ² − T·λ + D = 0, we have A² − T·A + D·I = 0 This allows computing A⁻¹: multiply by A⁻¹ to get A − T·I + D·A⁻¹ = 0 → A⁻¹ = (T·I − A)/D

Trace properties: tr(A + B) = tr(A) + tr(B) tr(kA) = k·tr(A) tr(AB) = tr(BA) — cyclic property

Rotation matrix: R(θ) = [[cos θ, −sin θ], [sin θ, cos θ]] — orthogonal, det = +1

• Remember: det(AB) = det(A)·det(B); det(A⁻¹) = 1/det(A); det(A^T) = det(A); eigenvalues satisfy |A − λI| = 0; trace = sum of eigenvalues; Cayley-Hamilton lets you find powers of matrix via its characteristic equation
• Previous years: “Find inverse of [[2,1],[3,2]] using adjoint method” [2023]; “For matrix A, A² = I. Find possible eigenvalues of A” [2024]; “Solve using Cramer’s rule: 2x + y = 5, x + 3y = 6” [2024]

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📊 JEE Main Exam Essentials

Detail	Value
Questions	90 (30 per subject)
Time	3 hours
Marks	300 (90 per subject)
Section	Physics (30), Chemistry (30), Mathematics (30)
Negative	−1 for wrong answer
Mode	Computer-based

🎯 High-Yield Topics for JEE Main Mathematics

• Calculus (Differentiation + Integration) — ~35 marks combined
• Coordinate Geometry (straight lines, circles, conics) — ~20 marks
• Algebra (Complex Numbers, Quadratics, P&C, Probability) — ~25 marks
• Trigonometry + Inverse Trigonometry — ~15 marks
• Vector + 3D — ~15 marks

📝 Previous Year Question Patterns

• Matrices & Determinants: 1 question per year, 4 marks
• Common patterns: inverse of 2×2, solving linear equations, determinant of 3×3, adjoint, eigenvalues
• Weight: medium frequency, medium difficulty

💡 Pro Tips

• For 2×2 inverse: swap diagonal elements, negate off-diagonal, divide by determinant
• Never swap rows during determinant calculation without adjusting sign
• For system of equations with parameter, use rank method — checking consistency is important
• Cayley-Hamilton is powerful for computing A⁻¹ and A^n without division
• A·adj(A) = adj(A)·A = det(A)·I (this is always true)
• For adjoint method: find cofactors, transpose, divide by det
• Characteristic equation |A − λI| = 0 gives eigenvalues — then use these to find eigenvectors

🔗 Official Resources

• NTA Official JEE Main
• JEE Main Syllabus PDF

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