Mathematics chapters
  1. 013D Geometry
  2. 02Application of Derivatives
  3. 03Area Under Curves
  4. 04Binomial Theorem
  5. 05Circles
  6. 06Complex Numbers
  7. 07Continuity & Differentiability
  8. 08Definite Integration
  9. 09Determinants
  10. 10Differential Equations
  11. 11Ellipse
  12. 12Hyperbola
  13. 13Indefinite Integration
  14. 14Inequalities & Modulus
  15. 15Inverse Trigonometric Functions
  16. 16Limits
  17. 17Logarithms
  18. 18Matrices
  19. 19Methods of Differentiation
  20. 20Number System & Basics
  21. 21Parabola
  22. 22Permutations & Combinations
  23. 23Probability
  24. 24Properties of Triangle
  25. 25Quadratic Equations
  26. 26Sequences & Series
  27. 27Sets, Relations & Functions
  28. 28Statistics
  29. 29Straight Lines
  30. 30Trigonometric Equations
  31. 31Trigonometric Ratios & Identities
  32. 32Vectors

Chapter 9 · Class 11-12 & JEE Main/Adv

Determinants

35 topics in 12 parts

Full lessonClass 12 Mathematics · Determinants Practice3 practice questions
Definition & Order of Determinants Minors & Cofactors Expansion of Determinants Sarrus Rule Properties of Determinants Row & Column Operations

Key ideas

+Y +Z +X O(0,0,0) Line L A(x₁, y₁, z₁) a P(x, y, z) r Parallel Vector b = aî + bĵ + ck̂ Cartesian Form: (x - x₁)/a = (y - y₁)/b = (z - z₁)/c = λ ⟺ General Point: (x₁ + λa, y₁ + λb, z₁ + λc)
Determinants

Analytical Comparison & Distinction Matrix

Analytical MethodStandard RepresentationParametric / Invariant Framework
Algebraic NatureExplicit equations relating spatial coordinatesParameterized using scalar parameters (λ, μ) or vectors
Symmetry UsageRelies on coordinate expansionExploits vector dot/cross products for rapid simplification
Computational EfficiencyDirect for intercept evaluationSuperior for angles, orthogonal projections, and distances
Common MistakeSign confusion in determinantsOmitting boundary condition checks

All 35 topics in Determinants

  1. Determinants Introduction

    • Definition & Order of Determinants
    • Minors & Cofactors
    • Elementary Expansion
    • Sarrus Rule of Expansion

  2. Properties of Determinants

    • Elementary Row & Column Operations
    • Factorization of Determinants
    • Polynomial Determinants
    • Determinant Equations
    • Determinants of Matrix Powers
    • Simplification of Determinants
    • Trigonometric Determinants
    • Optimization & Extreme Values

  3. Evaluation of Determinants

    • Direct Evaluation
    • Algebraic & Parameterized Determinants

  4. Use of Determinant in Coordinate Geometry

    • Area of a Triangle
    • Collinearity of Points
    • Condition of Concurrency of Three Lines
    • Condition of General Second Degree Equation

  5. Product of Two Determinants

    • Column by Column
    • Row by Column
    • Row by Row

  6. Differentiation of a Determinant

    • Derivative Evaluation

  7. Summation Operator on Determinants

    • Sequence & Series in Determinants
    • Summation of Determinants

  8. Special Determinants

    • Circulant & Symmetric Determinants
    • Structured Determinants

  9. System of Linear Equations

    • Cramer's Rule
    • Homogeneous Systems & Non-Trivial Solutions
    • Inconsistent & Infinite Solutions
    • Consistency & Number of Solutions

  10. Limit of Determinant

    • Limits Involving Determinants

  11. Integration of Determinant

    • Definite & Indefinite Integration

  12. Adjoint & Singular Matrices

    • Adjoint Matrix Properties
    • Iterated Adjoint
    • Singularity Conditions