How to Solve: Determinants (Properties, Cramer’s Rule, Differentiation/Integration of Determinants) – IIT JEE Guide

How to Solve: Determinants (Properties, Cramer’s Rule, Differentiation/Integration of Determinants) – IIT JEE Guide

Introduction Mastering determinants doesn’t just get you 4-6 marks in IIT JEE—it unlocks Cramer’s Rule for solving linear systems, area/volume calculations in coordinate geometry, and even differentiation tricks in calculus that save you 10+ minutes in the exam. One wrong sign, and your entire solution collapses. This guide ensures you never lose marks on determinants again.

What You Need To Know First

1. Matrix Basics – Know how to write a 2×2 or 3×3 matrix and identify rows/columns.
2. Basic Algebra – Expand expressions like (a + b)(c + d) and handle signs (+/-) carefully.
3. Partial Derivatives (for differentiation of determinants) – Know how to differentiate a function with respect to one variable while treating others as constants.

Key Vocabulary

Formulas To Know

1. Determinant of a 2×2 Matrix

Formula: For A = [[a, b], [c, d]], |A| = ad - bc.

Variables: - a, b, c, d = elements of the matrix.

MEMORISE THIS (Not given in JEE sheet).

2. Determinant of a 3×3 Matrix (Expansion by First Row)

Formula: For A = [[a, b, c], [d, e, f], [g, h, i]], |A| = a(ei - fh) - b(di - fg) + c(dh - eg).

Variables: - a, b, c, ... = elements of the matrix.

MEMORISE THIS (Not given in JEE sheet).

3. Properties of Determinants

MEMORISE ALL PROPERTIES (Not given in JEE sheet).

4. Cramer’s Rule

Formula: For the system:

a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃

Let D = |A| (determinant of coefficient matrix). Let D₁ = determinant of matrix where 1st column is replaced by [d₁, d₂, d₃]. Let D₂ = determinant of matrix where 2nd column is replaced by [d₁, d₂, d₃]. Let D₃ = determinant of matrix where 3rd column is replaced by [d₁, d₂, d₃].

Then:

x = D₁ / D,   y = D₂ / D,   z = D₃ / D

Condition: D ≠ 0 (otherwise, no unique solution).

MEMORISE THIS (Not given in JEE sheet).

5. Differentiation of Determinants

Formula: If A(t) = [[a(t), b(t)], [c(t), d(t)]], then:

d/dt |A(t)| = |[a'(t), b'(t)], [c(t), d(t)]| + |[a(t), b(t)], [c'(t), d'(t)]|

For 3×3, differentiate one row at a time and sum the determinants.

MEMORISE THIS (Not given in JEE sheet).

6. Integration of Determinants

Formula: If A(x) = [[a(x), b(x)], [c(x), d(x)]], then:

∫ |A(x)| dx = ∫ (a(x)d(x) - b(x)c(x)) dx

(Integrate term by term.)

MEMORISE THIS (Not given in JEE sheet).

Step-by-Step Method

How to Compute a Determinant (3×3 Matrix)

1. Choose a row/column (preferably with zeros to simplify).
2. For each element in the row/column:
3. Delete its row and column to get the minor.
4. Multiply the element by its cofactor ((-1)^(i+j) × minor).
5. Sum all these products to get the determinant.

How to Use Cramer’s Rule

1. Write the system in matrix form AX = B.
2. Compute D (determinant of coefficient matrix A).
3. If D = 0, stop (no unique solution).
4. Compute D₁, D₂, D₃ by replacing columns of A with B.
5. Solve for x, y, z using x = D₁/D, y = D₂/D, z = D₃/D.

How to Differentiate a Determinant

1. Write the determinant as a function of t.
2. Differentiate one row at a time (treat other rows as constants).
3. Sum the determinants of the differentiated rows.

How to Integrate a Determinant

1. Expand the determinant into a polynomial.
2. Integrate term by term.

Worked Examples

Example 1 – Basic (3×3 Determinant)

Problem: Find |A| where A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]].

Solution: 1. Expand by first row:
|A| = 1 × |[5, 6], [8, 9]| - 2 × |[4, 6], [7, 9]| + 3 × |[4, 5], [7, 8]| 2. Compute 2×2 determinants:
- |[5, 6], [8, 9]| = (5×9) - (6×8) = 45 - 48 = -3
- |[4, 6], [7, 9]| = (4×9) - (6×7) = 36 - 42 = -6
- |[4, 5], [7, 8]| = (4×8) - (5×7) = 32 - 35 = -3 3. Substitute back:
|A| = 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0

What we did and why: We expanded along the first row because it’s straightforward. The determinant is 0 because the rows are linearly dependent (R₃ = 2R₂ - R₁).

Example 2 – Medium (Cramer’s Rule)

Problem: Solve using Cramer’s Rule:

2x + y = 5
x - 3y = -12

Solution: 1. Write coefficient matrix A and constant matrix B:
A = [[2, 1], [1, -3]], B = [5, -12] 2. Compute D:
|A| = (2 × -3) - (1 × 1) = -6 - 1 = -7 3. Compute D₁ (replace 1st column with B):
D₁ = |[5, 1], [-12, -3]| = (5 × -3) - (1 × -12) = -15 + 12 = -3 4. Compute D₂ (replace 2nd column with B):
D₂ = |[2, 5], [1, -12]| = (2 × -12) - (5 × 1) = -24 - 5 = -29 5. Solve for x and y:
x = D₁/D = -3 / -7 = 3/7
y = D₂/D = -29 / -7 = 29/7

What we did and why: We used Cramer’s Rule because the system is 2×2 and D ≠ 0. Always check D first—if it’s zero, Cramer’s Rule fails.

Example 3 – Exam-Style (Differentiation of Determinant)

Problem: If A(t) = [[t, t²], [1, t³]], find d/dt |A(t)|.

Solution: 1. Compute |A(t)| first:
|A(t)| = (t × t³) - (t² × 1) = t⁴ - t² 2. Differentiate directly (easier here):
d/dt |A(t)| = 4t³ - 2t 3. Verify using row differentiation (for practice):
- Differentiate 1st row: |[1, 2t], [1, t³]| = (1 × t³) - (2t × 1) = t³ - 2t
- Differentiate 2nd row: |[t, t²], [0, 3t²]| = (t × 3t²) - (t² × 0) = 3t³
- Sum: (t³ - 2t) + 3t³ = 4t³ - 2t (matches direct differentiation).

What we did and why: We first expanded the determinant to simplify differentiation. The row-wise method is useful for larger matrices where expansion is tedious.

Common Mistakes

Exam Traps

1-Minute Recap (Night Before Exam)

"Listen up—this is your 60-second determinant survival guide for JEE.

1. For 2×2: |A| = ad - bc. Memorise it.
2. For 3×3: Expand along the row/column with the most zeros. Use + - + for signs.
3. Properties:
4. Swap rows → flip sign.
5. Identical rows → determinant = 0.
6. Add a multiple of one row to another → no change.
7. Cramer’s Rule:
8. Compute D first. If D = 0, stop—no solution.
9. Replace columns with constants to get D₁, D₂, D₃.
10. x = D₁/D, y = D₂/D.
11. Differentiation:
12. Differentiate one row at a time, sum the determinants.
13. Integration:
14. Expand first, then integrate term by term.
15. Exam traps:
16. Check for D = 0 before Cramer’s Rule.
17. Watch for row operations—they can change the determinant.
18. Never assume a matrix is non-singular—always verify.

You’ve got this. Go crush those determinants!