Determinants
Learn Determinants with notes, examples, and practice questions.
Chapter Overview
A determinant assigns a single number to a square matrix. It tells whether a matrix is invertible, measures area/volume scaling, and solves linear systems via Cramer's rule.
Expansion and Properties
2×2
|A| = ad − bc for [[a,b],[c,d]]
Row interchange
Swapping two rows multiplies |A| by −1
Zero row
|A| = 0
Proportional rows
|A| = 0
Row operation Rᵢ→Rᵢ+kRⱼ
|A| unchanged
Product
|AB| = |A|·|B|
Adjoint and Inverse
Inverse
A⁻¹ = adj(A) / |A| (|A| ≠ 0)
Check
AA⁻¹ = I
Adjoint of A: transpose of cofactor matrix. For invertible A: A⁻¹ = (1/|A|)·adj(A) when |A| ≠ 0.
Singular matrix: |A| = 0 — no inverse exists.
Area and Linear Systems
- Area of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃) = ½|det of coordinate matrix|
- If |A| = 0 for coefficient matrix, system may have infinite or no solutions
- Cramer's rule: xᵢ = |Aᵢ|/|A| when |A| ≠ 0
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: 3×3 Determinant
Evaluate |[[1,2,3],[0,1,4],[5,6,0]]|.
- 1Expand along row 1 or use row operations
- 2R₃→R₃−5R₁: simplifies to triangular form
- 3|A| = 1·(1·0−4·6) = −24 (after tracking operations)
Answer
−24 (verify with Sarrus or expansion)
Example 2: Inverse of 2×2
Find A⁻¹ for A = [[3,1],[5,2]].
- 1|A| = 6−5 = 1
- 2A⁻¹ = [[2,−1],[−5,3]]
Answer
A⁻¹ = [[2,−1],[−5,3]]
Example 3: Area
Area of triangle (0,0), (4,0), (0,3).
- 1½|det [[4,0],[0,3]]| = ½|12| = 6
Answer
6 sq. units
Key Points to Remember
- ✓ |A| = 0 ⇔ A is singular
- ✓ Use row operations to simplify — track sign changes
- ✓ adj(A)·A = |A|·I
Exam Tips
- • Expand along row/column with most zeros
- • Show row operations clearly when simplifying
- • Verify A·A⁻¹ = I for inverse problems
Formula Cheat Sheet
Quick reference — NCERT board exam formulas
Determinants
- 2 × 2:|A| = ad − bc for [[a,b],[c,d]]
- |A| = 0 if two rows/columns are identical or proportional
- If a row/column is a sum of two, |A| splits into sum of two determinants
Practice MCQs — Determinants
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
|[[2,1],[4,3]]| equals: