Engineering and Architecture
Management and Business Administration
Medicine and Allied Sciences
Law
Animation and Design
Media, Mass Communication and Journalism
Finance & Accounts
Computer Application and IT
Pharmacy
Hospitality and Tourism
Competition
School
Study Abroad
Arts, Commerce & Sciences
Learn
Online Courses and Certifications

JEE Main Form Correction 2025 - Date (Out), Procedure, Guidelines, Fee

Home
JEE Main
Study Material
Inverse Matrix - Practice Questions & MCQ

Inverse Matrix - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Syllabus
Eligibility Criteria
Application Process
Admit Card
Answer Key
Preparation Tips
Counselling Process
Cutoff
Mock Test
Result
Exam Analysis
Student Reactions
Question Paper
Dates
College Predictor
Rank Predictor
Accepting Colleges
Study Material

Quick Facts

Inverse of a Matrix, Properties of Inverse of a Matrix - Part 3 are considered the most difficult concepts.
Properties of Inverse of a Matrix - Part 1 are considered the most asked concepts.
22 Questions around this concept.

Solve by difficulty

Let A be any 3×3 invertible matrix. Then which one of the following is not always true ?

A

$adj\left ( A \right )=\left | A \right |.A^{-1}$

Correct Answer

Wrong Answer

B

$adj\left ( adj\left ( A \right ) \right )=\left | A \right |.A$

Correct Answer

Wrong Answer

C

$adj\left ( adj\left ( A \right ) \right )=\left | A \right |^{2}.\left ( adj\left ( A \right ) \right )^{-1}$

Correct Answer

Wrong Answer

D

$adj\left ( adj\left ( A \right ) \right )=\left | A \right |.\left ( adj\left ( A \right ) \right )^{-1}$

Correct Answer

Wrong Answer

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

A

If $det\; A=\pm 1,then\; A^{-1}$ need not exist

Correct Answer

Wrong Answer

B

If $det\; A=\pm 1,then\; A^{-1}$ exists but all its entries are not necessarily integers

Correct Answer

Wrong Answer

C

If $det\; A\neq \pm 1,then\; A^{-1}$ exists and all its entries are nonintegers

Correct Answer

Wrong Answer

D

If $det\; A=\pm 1,then\; A^{-1}$ exists and all its entries are integers

Correct Answer

Wrong Answer

If A is an 3 x 3 non - singular matrix such that AA' = A'A and B = A^-1 A' , then BB' equals :

A

B^-1

Correct Answer

Wrong Answer

B

(B^-1)'

Correct Answer

Wrong Answer

C

I + B

Correct Answer

Wrong Answer

D

I

Correct Answer

Wrong Answer

New: Direct link to apply for JEE Main 2025 registration for session 1

Also Check: Crack JEE Main 2025 - Join Our Free Crash Course Now!

JEE Main 2025: Sample Papers | Syllabus | Mock Tests | PYQs | Video Lectures

JEE Main 2025: Preparation Guide | High Scoring Topics | Study Plan 100 Days

Concepts Covered - 5

Inverse of a Matrix

Inverse of a Matrix

A non-singular square matrix A is said to be invertible if there exists a non-singular square matrix B such that

AB = I = BA

and the matrix B is called the inverse of matrix A. Clearly, B should also have the same order as A.

Hence, $\\\mathrm{A^{-1} = B \Leftrightarrow AB = \mathbb{I}_n = BA}$

Formula for A^-1

We know

$\\\\ \\\mathrm{A(adj A)=|A|\mathbb{I}_n} \\\text{Multiplying both sides by A}^{-1} \\\mathrm{\Rightarrow A^{-1}A (adj A)=A^{-1}\mathbb{I}_n|A|} \\\mathrm{\Rightarrow\mathbb{I}_n(adj A)=A^{-1}|A|\mathbb{I}_n} \,\,\,\,\quad (As \,\,A^{-1}.A= I)\\\mathrm{A^{-1}=\frac{adj A}{\left | A \right |}}$

Inverse of a 2 x 2 matrix

$\\\mathrm{Let\;A\;is\;a\;square\;matrix\;of\;order\;2}\\\mathrm{A=\begin{bmatrix} a &b \\ c & d \end{bmatrix}}\\\mathrm{Then,}\\\mathrm{A^{-1}=\begin{bmatrix} a &b \\ c & d \end{bmatrix}^{-1}=\frac{1}{ad-bc}\begin{bmatrix} d &-b \\- c & a \end{bmatrix}}$

Inverse of a Matrix of order 3 using adjoint

To compute the inverse of matrix A of order 3, first, check whether the matrix is singular or non-singular.

If the matrix is singular, then its inverse does not exist.

If the matrix in non-singular, then the following ar ethe steps to find the Inverse

$We\,\,use\,\,the\,\,formula\,\,A^{-1}= \frac{1}{|A|}.adj(A)$

Calculating the Matrix of Minors,
then turn that into the Matrix of Cofactors,
then take the transpose (These 3 steps give us the adjoint of matrix A)
multiply that by 1/|A|.

For example,

Let compute the inverse of matrix A,

$\mathrm{A=\begin{bmatrix} 1 &1 &2 \\ 1& 2 & 3\\ 3 &1 &1 \end{bmatrix}}$

First, find the determinant of A

$\\\mathrm{\left |A \right |=\begin{vmatrix} 1 &1 &2 \\ 1& 2 & 3\\ 3 &1 &1 \end{vmatrix}=1\cdot(2\times1-3\times1)-1\cdot(1\times1-3\times3)+2\cdot(1\times1-3\times2)} \\\mathrm{\left | A \right |=-3\neq0}\\\mathrm{\therefore A^{-1}\;exists}$

Now, find the minor of each element

$\\\mathrm{M_{11}=\begin{vmatrix} 2&3 \\1 &1 \end{vmatrix}=2\times 1-3\times1=-1}\\\\\mathrm{M_{12}=\begin{vmatrix} 1&3 \\3 &1 \end{vmatrix}=1\times 1-3\times3=-8}\\\\\mathrm{M_{13}=\begin{vmatrix} 1&2 \\3 &1 \end{vmatrix}=1\times 1-2\times3=-5}$

And here is the calculation for the whole matrix:

Minor matrix

$M=\begin{bmatrix}2\times \:1-3\times 1&1\times 1-3\times 3&1\times 1-2\times 3\\ \:1\times 1-2\times 1&1\times \:1-2\times 3&1\times 1-3\times \:1\\ \:1\times \:3-2\times 2&1\times 3-2\times \:1&1\times 2\:-1\times \:1\end{bmatrix}=\begin{bmatrix}-1&-8&-5\\ -1&-5&-2\\ -1&1&1\end{bmatrix}$

Now Cofactor of the given matrix

We need to change the sign of alternate cells, like this $\begin{bmatrix}+&-&+\\ \:-&+&-\\ \:+&-&+\end{bmatrix}$

So, Cofactor matrix C = $\begin{bmatrix}+\left(-1\right)&-\left(-8\right)&+\left(-5\right)\\ \:-\left(-1\right)&+\left(-5\right)&-\left(-2\right)\\ \:+\left(-1\right)&-\left(1\right)&+\left(1\right)\end{bmatrix}=\begin{bmatrix}-1&8&-5\\ \:1&-5&2\\ \:-1&-1&1\end{bmatrix}$

Now to find the adj A, take the transpose of matrix C

$Adj\:A=C'=\begin{bmatrix}-1&1&-1\\ \:\:8&-5&-1\\ \:\:-5&2&1\end{bmatrix}$

$\\\mathrm{Hence, A^{-1}=\frac{adj\;A}{|A|}}\\\mathrm{A^{-1}=\frac{1}{-3}\begin{bmatrix}-1&1&-1\\ \:\:8&-5&-1\\ \:\:-5&2&1\end{bmatrix} =\begin{bmatrix}-\frac{1}{-3}&\frac{1}{-3}&-\frac{1}{-3}\\ \:\:\frac{8}{-3}&-\frac{5}{-3}&-\frac{1}{-3}\\ \:\:-\frac{5}{-3}&\frac{2}{-3}&\frac{1}{-3}\end{bmatrix}}\\\mathrm{A^{-1}=\begin{bmatrix}\;\frac{1}{3}&-\frac{1}{3}&\frac{1}{3}\\ \:\:-\frac{8}{3}&\frac{5}{3}&\frac{1}{3}\\ \:\:\frac{5}{3}&-\frac{2}{3}&-\frac{1}{3}\end{bmatrix}}$

Properties of Inverse of a Matrix - Part 1

Properties of inverse of a matrix:

1. Inverse of a matrix is unique

Proof:

Let A be a square and non-singular matrix and let B and C be two inverses of matrix A

$\\\mathrm{AB= BA =\mathbb{I}_n \;\; (since \;B\; is\; inverse\; of\; A)} \\\mathrm{AC= CA =\mathbb{I}_n \;\; (since \;C\; is\; inverse\; of\; A)} \\\mathrm{now, } \;\;\mathrm{AB=\mathbb{I}_n} \\\mathrm{C(AB)=C\mathbb{I}_n\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left [ Multiplication\;by\;C \right ]} \\\mathrm{(CA)B=C\mathbb{I}_n\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left [ by\;associativity \right ]} \\\mathrm{\mathbb{I}_nB=C\mathbb{I}_n} \Rightarrow B=C$

Hence an invertible matrix has a unique inverse.

2. If A and B are invertible matrices of order n, then AB will also be invertible. and (AB)^-1 = B^-1A^-1.

Proof :

$\\\text{A and B are invertible matrices}\text{, so}\;|A|\neq 0\;and\;|B|\neq0\\\text{Hence},\;|A||B|\neq0\Rightarrow |AB|\neq0$

$\\\text{now,}\;(AB)(B^{-1}A^{-1})=A(BB^{-1})A^{-1}\;\;\;\;\;\;\;\;\;\;\text{[by\;associative\;law]}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=}A(I_n)A^{-1}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathrm{[\because BB^{-1}=I_n]}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=}AA^{-1}=I_n\\\text{also,}\;(B^{-1}A^{-1})(AB)=B^{-1}(A^{-1}A)B\;\;\;\;\;\;\;\;\;\;\text{[by\;associative\;law]}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=}B^{-1}(I_nB)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathrm{[\because A^{-1}A=I_n]}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=}B^{-1}B=I_n\\\text{Thus,}\;\;(AB)(B^{-1}A^{-1})=I_n=(B^{-1}A^{-1})(AB)\\\text{Hence,}\;\;(AB)^{-1}=B^{-1}A^{-1}$

Properties of Inverse of a Matrix - Part 2

Properties of Inverse of a Matrix

3. If A is an invertible matrix, then

(A')^-1= (A^-1) '

Proof: As A is an invertible matrix, so |A| ≠ 0 ⇒ |A' | ≠ 0. Hence, A'is also invertible.

$\\\mathrm{Now, \;\;\;AA^{-1}=\mathbb{I}_n=A^{-1}A} \\\text{Taking transpose of all three sides}\\\mathrm{\Rightarrow (AA^{-1})'=(\mathbb{I}_n)' = (A^{-1}A)'} \\\mathrm{\Rightarrow (A^{-1})'A'=\mathbb{I}=A'(A^{-1})'} \\\mathrm{(A')^{-1} = (A^{-1})'}$

4. Let A be an invertible matrix, then, (A^-1)^-1=A

Proof:

Let A be an invertible matrix of order n.

$\\As\,\,A.A^{-1}=I=A^{-1}.A\\\mathrm{\Rightarrow \;\;\;\;\;(A^{-1})^{-1}=A}$

5. Let A be an invertible matrix of order n and k is a natural number, then (A^k)^-1 = (A^-1)^k= A^-k

Proof:

$\\\mathrm{(A^k)^{-1}=(A\times A\times A\times \ldots\times A)^{-1}}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;=(A^{-1}\times A^{-1}\times A^{-1}\times \ldots\times A^{-1})}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;=(A^{-1})}^k$

Properties of Inverse of a Matrix - Part 3

Properties of Inverse of a Matrix

6. Let A be an invertible matrix of order n, then

$\\\mathrm{|A^{-1}| = \frac{1}{\left | A \right |}}$

Proof: ∵ A is invertible, then |A| ≠ 0.

$\\\mathrm{now, \;\;AA^{-1} =\mathbb{I}_n=A^{-1}A} \\\\\mathrm{\Rightarrow |AA^{-1}|=|\mathbb{I}_n|} \\\\\mathrm{\Rightarrow |A||A^{-1}| = 1} \\\\\mathrm{\Rightarrow|A^{-1}|= \frac{1}{\left | A \right |}}$

7. Inverse of a non-singular diagonal matrix is a diagonal matrix

$\\\text{if }A = \begin{bmatrix} a &0 & 0\\ 0 & b &0 \\ 0 &0 & c \end{bmatrix} \text{and}\;\mathrm{|A|\neq 0}$

then

$\mathrm{A^{-1}=\begin{bmatrix} \frac{1}{a} &0 & 0\\ 0& \frac{1}{b} &0 \\ 0& 0 & \frac{1}{c} \end{bmatrix}}$

Study it with Videos

Inverse of a Matrix

Inverse of a Matrix of order 3 using adjoint

Properties of Inverse of a Matrix - Part 1

Properties of Inverse of a Matrix - Part 2

Properties of Inverse of a Matrix - Part 3

Study other Related Concepts

Inverse Matrix Current Topic

Determine The Order Of Matrix

Types of Matrices

Triangular matrix (Upper and Lower triangular matrix)

Matrix operations

Matrix Multiplication

Transpose of a Matrix

Symmetric Matrix & Skew Symmetric Matrix

Conjugate of a Matrix

Hermitian matrix

Skew Hermitian Matrix

"Stay in the loop. Receive exam news, study resources, and expert advice!"

Get Answer to all your questions

JEE Main Articles

How to Check JEE Mains Exam Centre

Nov 18, 2024

JEE Main Result 2025 - Date, BE, BTech, BArch, BPlan Results Link

Nov 18, 2024

JEE Main 2025 Latest News and Updates Today

Nov 18, 2024

JEE Mains 2025 - Registration (Open), Exam Dates Out, Syllabus, Pattern, Previous Year Papers

Nov 18, 2024

IIT Kanpur JEE Main 2025 Preparation - 45 Days Days Crash Course SATHEE

Nov 18, 2024

Lowest JEE Main Registration this Year? - Students Face Issues in Application, Check Trends

Nov 18, 2024

Upcoming Engineering Exams 2025 - Latest Updates on JEE, VITEEE, BITSAT, MHT CET, State Exams

Nov 18, 2024

How many Students Appeared for JEE Main 2025?

Nov 18, 2024

Quick Facts
Practice Questions
Concept Covered
Concept Video

B.Tech/B.Arch Admissions OPEN

UPES B.Tech Admissions 2025

Ranked #42 among Engineering colleges in India by NIRF | Highest CTC 50 LPA , 100% Placements

Dayananda Sagar University B.Tech Admissions 2025

60+ Years of Education Legacy | UGC & AICTE Approved | Prestigious Scholarship Worth 6 Crores | H-CTC 35 LPA

Parul University B.Tech Admissions 2025

India's youngest NAAC A++ accredited University | NIRF rank band 151-200 | 2200 Recruiters | 45.98 Lakhs Highest Package

Lovely Professional University B.Tech Admissions 2025

India's Largest University | NAAC A++ Accredited | 100% Placement Record | Highest Package Offered : 3 Cr PA

Atlas SkillTech University | B.Tech Admissions

Hands on Mentoring and Code Coaching | Cutting Edge Curriculum with Real World Application

Pearson | PTE

Register now for PTE & Unlock 20% OFF : Use promo code: 'C360SPL20'. Valid till 30th NOV'24! Trusted by 3,500+ universities globally

View All Application Forms

News and Notifications

November 05, 2024 - 04:02 PM IST :

NTA has announced the JEE Main exam date 2025 for sessions 1 and 2.

October 29, 2024 - 11:15 AM IST :

JEE Main 2025 registration for session 1 started at jeemain.nta.nic.in.

BTech Admission 2025 Live: JEE Mains, SRMJEEE, MET registrations underway; MHT CET syllabus updates

November 17, 2024 - 09:11 PM IST

JEE Mains 2025 session 1 registration deadline on November 22; apply on jeemain.nta.nic.in

November 17, 2024 - 03:28 PM IST

BTech Admissions 2025 Live: JEE Mains, SRMJEE, VITEEE dates out; AEEE, MET registrations underway

November 15, 2024 - 11:12 AM IST

Centre issues coaching guidelines, prohibits centres from using false claims like '100% selection'

November 13, 2024 - 06:45 PM IST

BTech Admissions 2025: JEE Mains registration underway; updates on state engineering exams

November 13, 2024 - 11:45 AM IST

BTech Admissions 2025: JEE Mains, VITEEE, SRMJEE exam dates announced

November 12, 2024 - 10:27 PM IST

BTech Admissions 2025: JEE Main registration last date, direct link

November 07, 2024 - 10:09 PM IST

Explore on Careers360

JEE Main

JEE Main Application Form

JEE Main Eligibility Criteria

JEE Main Admit card

JEE Main Syllabus

JEE Main Exam Pattern

JEE Main Answer Key

JEE Main Cutoff

JEE Main Result

JEE Advanced

JEE Advanced Application Form

JEE Advanced Eligibility Criteria

JEE Advanced Admit card

JEE Advanced Syllabus

JEE Advanced Exam Pattern

JEE Advanced Answer Key

JEE Advanced Cutoff

JEE Advanced Result

AP EAMCET

AP EAMCET Application Form

AP EAMCET Eligibility Criteria

AP EAMCET Admit card

AP EAMCET Syllabus

AP EAMCET Exam Pattern

AP EAMCET Answer Key

AP EAMCET Cutoff

AP EAMCET Result

TS EAMCET

TS EAMCET Application Form

TS EAMCET Eligibility Criteria

TS EAMCET Admit card

TS EAMCET Syllabus

TS EAMCET Exam Pattern

TS EAMCET Answer Key

TS EAMCET Cutoff

TS EAMCET Result

MHT CET

MHT CET Application Form

MHT CET Eligibility Criteria

MHT CET Admit card

MHT CET Syllabus

MHT CET Exam Pattern

MHT CET Answer Key

MHT CET Cutoff

MHT CET Result

KCET

KCET Application Form

KCET Eligibility Criteria

KCET Admit card

KCET Syllabus

KCET Exam Pattern

KCET Answer Key

KCET Cutoff

KCET Result

VITEEE

VITEEE Application Form

VITEEE Eligibility Criteria

VITEEE Admit card

VITEEE Syllabus

VITEEE Exam Pattern

VITEEE Answer Key

VITEEE Cutoff

VITEEE Result

BITSAT

BITSAT Application Form

BITSAT Eligibility Criteria

BITSAT Admit card

BITSAT Syllabus

BITSAT Exam Pattern

BITSAT Answer Key

BITSAT Cutoff

BITSAT Result

Colleges By Degree

BE/B.Tech Colleges in India

B.Arch Colleges in India

Dual Degree Colleges in India

Colleges By Branches

Computer Science Engineering Colleges in India

Automobile Engineering Colleges in India

Mechanical Engineering Colleges in India

Civil Engineering Colleges in India

Electrical Engineering Colleges in India

Information Technology Colleges in India

Electronics and Communication Engineering Colleges in India

Electrical and Electronics Engineering Colleges in India

Colleges By Exam

Engineering Colleges in India Accepting JEE Main

Engineering Colleges in India Accepting MHT CET

Engineering Colleges in India Accepting AP EAMCET

Engineering Colleges in India Accepting NATA

Engineering Colleges in India Accepting TS EAMCET

Engineering Colleges in India Accepting KCET

Engineering Colleges in India Accepting KEAM

Engineering Colleges in India Accepting BITSAT

By City

Engineering Colleges in Bengaluru

Engineering Colleges in Pune

Engineering Colleges in Hyderabad

Engineering Colleges in Chennai

Engineering Colleges in Mumbai

Engineering Colleges in Coimbatore

Engineering Colleges in Kolkata

Engineering Colleges in Delhi

By State

Engineering Colleges in Maharashtra

Engineering Colleges in Karnataka

Engineering Colleges in Telangana

Engineering Colleges in Andhra Pradesh

Engineering Colleges in Madhya Pradesh

Engineering Colleges in Uttar Pradesh

Engineering Colleges in Tamilnadu

BE/B.Tech

BE/BTech in Mechanical Engineering Colleges in India

BE/BTech in Computer Science Engineering Colleges in India

BE/BTech in Electronics and Communication Colleges in India

BE/BTech in Civil Engineering Colleges in India

BE/BTech in Electrical and Electronics Engineering Colleges in India

BE/BTech in Information Technology Colleges in India

BE/BTech in Electrical Engineering Colleges in India

BE/BTech in Chemical Engineering Colleges in India

Diploma

Diploma in Engineering Colleges in India

Diploma in Chemical Engineering Colleges in India

Diploma in Mechanical Engineering Colleges in India

Diploma in Computer Science Engineering Colleges in India

Diploma in Electronics and Communication Colleges in India

Diploma in Civil Engineering Colleges in India

Diploma in Electrical and Electronics Engineering Colleges in India

Diploma in Information Technology Colleges in India

Download Careers360 App

All this at the convenience of your phone

Regular Exam Updates
Best College Recommendations
College & Rank predictors
Detailed Books and Sample Papers
Question and Answers

Scan and download the app

OR

About
Hiring
Magazine
News
Contact
Blogs

Under Graduate
Post Graduate
JEE Main 2025
GATE 2025
SRMJEEE 2025
TS EAMCET 2025
BITSAT 2025
KCET 2025
MET 2025
MHT CET 2025
JEE Advanced 2025
AP EAMCET 2025

Top Engineering Colleges in India
Compare Colleges
Top Engineering College Reviews
Engineering Colleges in India
List of Top IITs in India
List of Top NITs in India
Engineering Admission Form
Top Government Engineering Colleges in India
List of Top IIITs in India
Top Private Engineering Colleges in India
Top Government Engineering Colleges in Hyderabad
Top Government Engineering Colleges in New Delhi
Top Government Engineering Colleges in Kolkata
IIT Delhi

JEE Advanced College Predictor
JEE Main College Predictor
JEE Main Rank Predictor
GATE College Predictor with PSU Chances
JEE Advanced Rank Predictor
UPCET College Predictor
MHT CET College Predictor
AP EAMCET College Predictor
JEE Main Architecture College Predictor
KCET College Predictor
GUJCET College Predictor
COMEDK UGET College Predictor
Engineering E-books
Sample Papers for Engineering
TS EAMCET College Predictor

B. Tech Companion
JEE Main Study Material
JEE main Online Preparation
Download E-Books and Sample Papers
Popular Engineering Courses
NCERT
NCERT Solutions
Careers360 Youtube Channel
Counselling Webinars
Download App
QnA - Get answers to your doubts
JEE Main 2025 Updates
JEE Main 2025 Syllabus
NIRF Ranking 2024 Engineering Colleges
JEE Main Previous Year Question Papers
Last 15 Years GATE Question Papers with Solutions

Sitemap
Disclaimer
Comment Policy
Privacy Policy
Grievance Redressal

Copyright © 2024 Pathfinder Publishing Pvt Ltd.

Joint Entrance Examination (Main)

JEE Main 2025 Sample Papers

Back to top