Matrices and Determinants

Board: JEE Main |Exam:JEEMain

JEEMainstudynotesforMatricesandDeterminantsbyAjayYadav.ThischaptercoversallconceptsrequiredforJEEMainwithproblem-solvingtechniques.

KeyConcepts

MatrixOperations

Multiplication,transpose,trace.Properties:(AB)’=B’A’.(ABC)⁻¹=C⁻¹B⁻¹A⁻¹.

AdjointandInverse

A(adjA)=(adjA)A=|A|I.A⁻¹=adjA/|A|. |A⁻¹|=1/|A|.(AB)⁻¹=B⁻¹A⁻¹.

PropertiesofDeterminants

|A’|=|A|.Row/columninterchangeflipssign.Identicalrows/cols⇒|A|=0. |kA|=kⁿ|A|. |AB|=|A||B|.

System of Equations

Cramer’s rule: x_j=|A_j|/|A|. Consistency: |A|≠0 ⇒ unique. |A|=0 and (adjA)B=0 ⇒ infinite. |A|=0 and (adjA)B≠0 ⇒ no solution.

Important Formulas

|kA|	kⁿ|A| for n×n matrix
|AB|	|A||B|
A⁻¹	adjA/|A|
Cramer’s rule	xj = |Aj|/|A|

Solved Examples

Example 1: For A=(1234), find |A|, adjA, A⁻¹.

Solution: |A|=-2. adjA=(4-2-31). A⁻¹=(-211.5-0.5).

Example 2: Test consistency: x+2y=5, 3x+6y=15.

Solution: |A|=6-6=0. (adjA)B=(6-2-31)(515)=(00). Infinite solutions.

Practice Questions

1. Find |A| for A=(123456789).
2. Find A⁻¹ for A=(1235).
3. Solve: 2x+3y=1, x-y=3 using Cramer’s rule.
4. If A=(cos&theta-sin&thetasin&thetacos&theta), show A’A=I.
5. Find conditions for k if system has infinite solutions: x+2y=5, 3x+ky=15.

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