JEE Maths Determinants | Formulas, Properties & Applications – StudentBro

Overview of Determinants for JEE Maths

Determinants is a core chapter of Class 12 Mathematics and is highly relevant in JEE Mathematics. This topic provides a systematic way to solve linear equations, analyze matrices, and understand properties of determinants. Questions from determinants appear frequently in both JEE Main and Advanced, often in formula-based and problem-solving formats.

Mastering this chapter ensures students can tackle linear algebra problems efficiently and score high marks in JEE.

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Introduction to Determinants

A determinant is a scalar value associated with a square matrix, which helps in solving linear equations and evaluating properties of matrices. Determinants are defined only for square matrices of order 1, 2, 3, and higher.

In JEE, understanding evaluation, expansion, and properties of determinants is key to solving complex problems quickly.

Determinant of a 2×2 Matrix

For a 2×2 matrix:

A = [a b
     c d]
det(A) = ad - bc
    

This is the simplest and most frequently tested determinant in JEE Main exams.

Determinant of a 3×3 Matrix

For a 3×3 matrix:

A = [a b c
     d e f
     g h i]
det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
    

JEE Advanced often tests cofactor expansion and properties for 3×3 determinants.

Minor and Cofactor

• Minor (M_ij): Determinant obtained by removing the i-th row and j-th column.
• Cofactor (C_ij): C_ij = (-1)^i+j M_ij

Cofactors are widely used in determinant expansion and inverse of a matrix.

Properties of Determinants

• Row/column swapping changes sign of determinant.
• Multiplying a row/column by a scalar multiplies determinant by that scalar.
• Determinant is zero if two rows/columns are identical or proportional.
• Adding a multiple of one row/column to another does not change determinant.
• Determinant of a triangular matrix = product of diagonal elements.

Expansion of Determinants

Determinants can be expanded along any row or column using cofactors. Formula for expansion along i-th row:

det(A) = a_i1*C_i1 + a_i2*C_i2 + ... + a_in*C_in

Applications of Determinants in JEE

• Solving linear equations using Cramer’s Rule.
• Finding area of a triangle given coordinates of vertices:

  Area = 1/2 * | x1 y1 1
                 x2 y2 1
                 x3 y3 1 |
              

• Finding volume of a tetrahedron in 3D geometry:

  V = 1/6 * | x1 y1 z1 1
               x2 y2 z2 1
               x3 y3 z3 1
               x4 y4 z4 1 |
              

Cramer’s Rule

Cramer’s Rule uses determinants to solve a system of n linear equations with n variables:

x_i = det(A_i) / det(A),  det(A) ≠ 0
    

Where A is the coefficient matrix and A_i is obtained by replacing the i-th column with constants. This method is frequently tested in JEE Main for 2×2 and 3×3 systems.

Determinant Identities

• det(AB) = det(A) * det(B)
• det(A^T) = det(A)
• det(kA) = kⁿ * det(A) for n×n matrix
• det(A^-1) = 1 / det(A)

Common Mistakes Students Make

• Forgetting signs in cofactor expansion.
• Ignoring properties of determinants to simplify.
• Errors in Cramer’s Rule calculations.
• Using 2×2 formula for 3×3 problems.

Importance of Determinants in JEE

• Have consistent weightage in both Main and Advanced exams.
• Support linear equations, matrices, and geometry.
• Test algebraic manipulation and problem-solving skills.
• Are formula-based, making them scoring if practiced regularly.

Preparation Tips for JEE Aspirants

• Memorize formulas and properties of determinants.
• Practice expansion and simplification techniques.
• Solve linear equation problems using Cramer’s Rule.
• Regularly attempt previous year JEE questions for pattern recognition.

Connection with Other Chapters

• Matrices – determinant is fundamental in matrix operations.
• Linear Equations – Cramer’s Rule is based on determinants.
• Geometry – area and volume problems.
• 3D Geometry – volume of tetrahedrons.

Conclusion

Determinants is a conceptual and highly scoring chapter in JEE Mathematics. With clear understanding of properties, formulas, and applications, students can tackle both simple and complex problems effectively. StudentBro provides exam-focused explanations, step-by-step strategies, and structured learning resources to help aspirants excel in JEE Main and Advanced.