📚 Key Concepts
Determinant Definition- Value associated with square matrices
- Denoted as |A| or det(A)
- 2×2 determinant: |a₁₁ a₁₂| = a₁₁a₂₂ - a₁₂a₂₁|a₂₁ a₂₂|
- 3×3 uses cofactor expansion
Key Terms- Minor (Mᵢⱼ): Determinant after removing iᵗʰ row and jᵗʰ column
- Cofactor (Cᵢⱼ): Cᵢⱼ = (-1)ᵢ⁺ʲMᵢⱼ
- Adjoint: Transpose of cofactor matrix
- Area of triangle = ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|
🎯 Key Formulas
|AB| = |A| × |B|For inverse: A⁻¹ = (adj A)/|A| if |A| ≠ 0Properties:- |kA| = k ⁿ|A| for n×n matrix
- |Aᵀ| = |A|
- |A⁻¹| = 1/|A|
⚠️ Common Mistakes to Avoid
Wrong sign in cofactor calculationIncorrect expansion orderForgetting determinant propertiesWrong adjoint calculationIncorrectly applying area formulaAssuming non-square matrices have determinants
📖 Knowledge Prerequisites
Matrix operationsBasic algebraProperties of 2×2 matricesUnderstanding of inverse matrices
💡 Tips for Students
Learn 2×2 determinant formula thoroughlyUse cofactor expansion systematicallyPractice step-by-step solutionsRemember: only square matrices have determinantsCheck for special cases (triangular, diagonal matrices)
👉 Practice Recommedations
Calculate 2×2 and 3×3 determinantsFind minors and cofactorsCalculate adjointsFind inverse using determinantsSolve system of equations using Cramer's ruleArea problems using determinants
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