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Cheat sheet · JEE Preparation Mathematics

Determinants cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

JEE · Mathematics

Cheat sheet

Class 12 · Chapter 4Determinants

A⁻¹ = adj A / |A| — only when |A| ≠ 0.

1. Basics

∣abcd∣=ad−bc\begin{vmatrix}a & b\\ c & d\end{vmatrix} = ad - bc
  • Minor Mᵢⱼ; cofactor Aᵢⱼ = (−1)ⁱ⁺ʲ Mᵢⱼ.
  • Expand along any row or column.

2. Area of a Triangle

Δ=12∣x1y11x2y21x3y31∣\Delta = \dfrac12\begin{vmatrix}x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1\end{vmatrix}

Take the absolute value; Δ = 0 ⇒ points collinear.

3. Adjoint & Inverse

A(adjA)=∣A∣IA(\text{adj}A) = |A|IA−1=1∣A∣adjAA^{-1} = \dfrac{1}{|A|}\text{adj}A
∣AB∣=∣A∣∣B∣|AB| = |A||B|∣kA∣=kn∣A∣|kA| = k^n|A|∣adjA∣=∣A∣n−1|\text{adj}A| = |A|^{n - 1}

4. System of Equations (AX = B)

ConditionSolution
|A| ≠ 0unique: X=A−1BX = A^{-1}B
|A| = 0, (adj A)B ≠ Ono solution (inconsistent)
|A| = 0, (adj A)B = Oinfinitely many or none

5. Key Points

  • Singular matrix: |A| = 0.
  • |A′| = |A|.
  • Swapping rows changes the sign.
  • Two identical rows ⇒ determinant 0.

Remember: Cofactors | area formula | A⁻¹ = adj A/|A| | consistency

Determinants: questions and notes

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