JEE MAIN 2021: MATRICES & DETERMINANTS in ONE SHOT

JEE MAIN 2021: MATRICES & DETERMINANTS in ONE SHOT

JEE MAIN 2021: CRASH COURSE MATRICES &

DETERMINANTS in ONE SHOT


#LetsKillJEE

LetsKillJEE


Chapter name 7th Jan-I

7th Jan-II

8th Jan-I

8th Jan-II

9th Jan-I

9th Jan-II

Matrices & Determinants 2 2 2 2 1 2

JEE MAIN 2020

Chapter name2nd Sept

-I

2nd Sept-II

3rd Sept

-I

3rd Sept-II

4th Sept

-I

4th Sept-II

5th Sept

-I

5th Sept-II

6th Sept

-I

6th Sept-II

Matrices & Determinants 1 2 2 2 2 2 2 2 2 2


Chapter name 8th Apr(I)

8th Apr(II)

9th April(I)

9th April(II)

10th April(I)

10th April(II)

12th Apr(I)

12th April(II)

Matrices & Determinants

1 2 2 2 2 2 2 1

Chapter name 9th jan(I)

9th jan(II)

10th jan(I)

10th jan(II)

11th jan(I)

11th jan(II)

12th jan(I)

12th jan(II)

Matrices & Determinants

2 2 1 2 2 2 2 2

JEE MAIN 2019


Chapter name 2018 2017 2016 2015

Matrices & Determinants 2 2 2 2

Live Interactive Classes● LIVE & Interactive Teaching Style

● Fun Visualizations, Quizzes &

Leaderboards

● Daily Classes on patented WAVE platform

● Dual Teacher Model For Personal

Attention

Why Vedantu Pro is the Best?Test Series & Analysis

● Exclusive Tests To Enable Practical Application

● Mock & Subject Wise Tests, Previous Year Tests

● Result Analysis To Ensure Thorough Preparation

Assignments & Notes

● Regular Assignments To Ensure Progress

● Replays, Class Notes & Study Materials Available 24*7

● Important Textbook Solutions

● Full syllabus Structured Long Term Course

● 20+ Teachers with 5+ years of experience

● Tests & Assignments with 10,000+ questions

● Option to Learn in English or Hindi

● LIVE chapter wise course for revision

● 4000+ Hours of LIVE Online Teaching

● Crash Course and Test Series before Exam

What is Included in Vedantu Pro?

₹7,200/-

₹20,400/-

₹40,800/-

NAGPRO

20% *Link available in description

(7,500)@ ₹6000/Month

(21,000)@ ₹5,600/Month

(39,000)@ ₹5,200/Month

20% *JEE Pro Subscription Link available in description

Apply Coupon Code: NAGPRO

How to Avail The Vedantu Pro Subscription

*Vedantu Pro Subscription Link available in description

Use the code: NAGPRO

1Select Your

Grade & Target

Click on Get Subscription

2 3Apply Coupon Code and Make Payment

Choose Your Subscription & Click on Proceed To Pay

4



jth column

ith row

[aij] m×n

rows

columns

A Matrix is an arrangement of various elements

m×n

ai

j

a11 a12 …………… a1na21 a2

2

…………… a2

n

….

am

1

am

2

…………… am

n

…. Elements are represented,

in the same way as we did in determinants

Matrices


Types of Matrices

1) Row matrix : A matrix with only one row and plural columns.

A matrix with only one column and plural rows.

1 × na11 a12. . . . . . . . a1n

m × 1

a11a21

am1

. . . . . .

2) Column matrix :


Matrix in which all the elements are Zero.

i.e. aij = 0, ∀ i, j

m × n

0 0 …………… 00 0 …………… 0…

.

0 0 …………… 0

….

….

3) Zero/Null matrix :

4) Horizontal matrix: A Matrix that looks horizontal. i.e. has More Columns and Less Rows or n > m

m × n

Types of Matrices


5) Vertical matrix: A Matrix that looks vertical. i.e. has More Rows and Less Columns or m > n

m x n

What are different types of matrices?

Types of Matrices


Types of Matrices

A matrix that has Equal number of rows and columns.

i.e. m = n

m×n

aii

a11 a12 a1n

a21 a2

2

a2

n….

an1 an2 …...… an

n

….

…...……...…

In a Square matrix, the elements with

same column and row number

are called Diagonal Elements.

i.e. elements with i = jFor Square matrices:

6) Square matrix :


Trace : Trace is the Sum of Diagonal Elements.

∑aii

n

i = 1= a11+ a22 + a33 ….. and so on


A) Triangular Matrix A Matrix with either, all upper diagonal or lower diagonal entries ZERO

a11 a12 a1n

a2

2

a2n

am

n

….

…...……...…a13

a23a33 a3n…...…

0

….

0 0 …...…

0 0

0

I) Upper Triangular Matrix Matrix with, all lower diagonal entries: Zero i.e. aij = 0, if ‘i > j’



II) Lower Triangular Matrix: Matrix with, all upper diagonal entries Zeroi.e. aij = 0, if ‘i < j’

a11

a21 a22

….

an1 an2 …...… amn

a33a31 a32

am3

0 00

….

…...……...…0

00…...…



III) Diagonal Matrix: Matrix with, all non-diagonal entries: Zeroi.e. aij = 0, if ‘i ≠ j’

0 00 0

….

0 0 …...…

….

…...……...…0

00 0 0…...…

0

a11

a2

2

amn

a33

What is triangular matrix and define various types of triangular matrices ?


Algebra of Matrices

Two matrices are said to be equal if:

1. Their orders are equal

2. Corresponding elements are equal.

⇒ 2x = 42x

2

1

3=

4

2

1

3

Equality:

Example:

How are calculations performed in matrices?

= ⇒ has no solution

as a21 is not equal

4

1

3

2

2x

2

x + 1

2


Addition of matrices Two matrices can be added only if :

1. order of the matrices are equal2. are to be added only term by term

Algebra of Matrices

Am×n + Bm×n = Cm×n

then, aij + bij = cij

Example:

1+2

1+(–1)

2+4

0+2

3+0

1+0+

1

1

2

0

3

1

2

–1

4

2

0

02×3 2×3

=

2×3

=30

62

31

2×3


Algebra of Matrices

=

Similarly while taking common out it will

come out from all entries

Multiplication by a constant

a11a21 a22

a33a31 a32

a12 a13a23k × =

ka11ka21 ka22

ka33ka31 ka32

ka12 ka13ka23

When a matrix is multiplied by aconstant k, then every element of

the matrix gets multiplied by k

211

420

2 ×422

840

Example


Algebra of Matrices

Multiplication of two Matrices : This is similar to multiplication of two

determinants.

A B C=

Post multiplier

Pre multiplier

×m× n n × p m×p

Number of Columns of First Matrix should be equal to number

of Rows of Second Matrix .

So, for multiplication of two Matrices


A × B ≠ B × A

1. Matrix multiplication is not commutative in general.

Not always true

(A×B) × C = A × (B×C)

2. Matrix multiplication is associative.

3. Matrix multiplication is distributive

A × (B + C) = A×B + A×C

Properties:


Properties:

4. If you multiply any matrix by null matrix, you get a null matrix

Am×n × On×n = Om×n

5. Exponential laws hold for square matrices, i.e.

An = A × An-1

Only for square matricesAm × An = Am+n

(Am)n = Amn


Note:

All the following definitions are defined ONLY for Square Matrices.

Symmetric AT = A

Idempotent A2 = A

Nilpotent A2 = On

Skew–Symmetric AT = –A

Involutory A2 = I

Orthogonal A×AT = I


Involutory Matrix :

Matrix A is said to be involutory if A2 = I

Properties

i) An = I n is an Even Integer

ii) Am = A m is an Odd Integer

iii) |A | = ± 1


Idempotent Matrix:

Matrix A is said to be idempotent if A2 = A

Properties

i) An = A ∀ n ≥ 2

ii) |A | = 0,1


Nilpotent Matrix

Matrix A is said to be Nilpotent Matrix of order ‘m’ if

Am = 0 and Am–1 ≠ 0

Properties

i) An = On ∀ n ≥ m

A = ⇒ A2 = 0 Order = 2

A is a Nilpotent Matrix of Order : 2

0 10 0

Example


Transpose of a matrix

Matrix obtained by Interchanging Rows and Columns is called Transpose of a Matrix.

Symbol : AT or A′

AIf =a1

a2

a3

b1

b2

b33 × 2

⇒ AT =a1 a2 a3

b1 b2 b3 2 × 3


1) (AT)T = A

2) (A + B)T = AT + BT

3) (KA)T

= K(AT)

(Reversal law holds) 4) (AB)T

= BTAT

Add and transpose is same as transpose and add

(A1 × A2…. An)T = (AnT ….. A2

T A1T)

K → constant

Generalized as

Properties :


Orthogonal matrix :

Matrix A is said to be an Orthogonal Matrix if:

Properties If A is an Orthogonal Matrix then,

A×AT = I

|A| = ±1


Matrix A is said to be a Symmetric Matrix if:

i.e. aij = aji

Symmetric Matrix:

AT = A

which means it basically is Symmetric about the Diagonal.

Exampleabc f g

e fb c


Matrix A is said to be Skew–Symmetric if:

i.e. aij = – aji

Skew – Symmetric Matrix:

AT = –A

Example0

–b–c –f 0

0 fb c

aii = 0 ∀ i ≤ n

Proof: aij = –aji

⇒ aii = –aii⇒ aii = 0

Properties


Every matrix A can be written as sum of a Symmetric and a Skew–Symmetric Matrix.

A = 12 (A + AT) + 1

2 (A – AT)

Symmetric Skew – symmetric

The determinant of Skew-Symmetric Matrix of Odd Order is Zero.


Adjoint of a Matrix

For any Square Matrix A, Adjoint of ‘A’ is defined as ‘Transpose’ of the Matrix obtained by replacing all elements of Matrix A by their

“Co–Factors”.A

=a11 a12 a13a21 a22 a23a31 a32 a33

C11 C12 C13C21 C22 C23C31 C32 C33

=

Co factor Matrix of A

Adj(A) = (Co – factor Matrix (A))T

So Adj (A) = C11 C21 C31C12 C22 C32C13 C23 C33


Properties of Adjoint of a Matrix

For Non–Singular Square Matrices, A and B we have the following results:

adj (AB) = adj(B)×adj(A)

adj(AT) = (adj A)T

|adj(A)| = |A|n–1

adj(adj(A)) = |A|n–2 × A

|adj(adj(A))| = |A|(n–1)2

adj(kAn) = kn–1 adj (An)

(Reversal law holds)

(k ⇒ constant)

(AT⇒ Transpose)

(|A|⇒ Determinant)

Property 1

Property 2

Property 3

Property 4

Property 5

Property 6


The conjugate of a complex number is denoted by Z

If Z = a + ib then Z = a – ib

Clearly if A is non-singular (i.e. |A| ≠ 0) then A–1 is defined, and is given by

A–1 = 1

|A|× (adj(A)) Note: A–1 × A = I

Inverse of Matrix

A square matrix of order n is invertible if there exists a square matrix B of the same order such that,

AB = In = BA

In such a case, we say that the inverse of A is B and we write, A–1 = B.


Properties of Inverse of a Matrix

Property 1

Property 2 (AB)–1 = B–1A–1

(AT)–1 = (A–1)T

Inverse of a matrix is unique

(Reversal law)

Property 3If A is an invertible square matrix; Then (A)T is also invertible and

Property 4 The inverse of an invertible symmetric matrix is a symmetric matrix.

Property 5 | A–1 | = | A |–1 i.e. | A–1 | =1

| A |


Homogeneous Equations

Ax =0

|A|≠0Trivial Solutions

x⁼y ⁼ z ⁼ 0

|A| =0

Infinite Solutions(trivial) & nontrivial

x, y, z not then all O’s for consistency


CRAMER’S RULE

D ≠ 0 D = 0

At least One D1,D2,D3≠ 0Consistent

Unique non - zero trivial

solution

D1=D2=D3 = 0Consistent Infinitely Solutions

(except // lines) no solution

D1=D2=D3 = 0Consistent

trivial solution

One of D1,D2,D3≠ 0 Inconsistent

solution


#LetsKillJEE

LetsKillJEE

JEE MAIN 2021 CRASH COURSE Course Overview

1. Cover entire JEE Main Syllabus (as per the new pattern) with India’s Best Teachers in 90 Sessions

2. Solve unlimited doubts with Doubt experts on our Doubt App from 8 AM to 11 PM

3. 10 full and 10 Part syllabus test to make you exam ready

4. Amazing tricks & tips to crack JEE Main 2021 Questions in power-packed 90 Min sessions

5. Learn on a 2-way Interactive Platform where the teacher is always with you

JEE MAIN 2021 CRASH COURSELightning Deal: ₹10000 - ₹8000/-

Batch Starts From : 16th Nov 2020

*Crash Course Link Available in Description

Apply Coupon Code: NAGCC


Q1.

A

B

12

4

D

C

10

-8

A

B

D

C

D

JEE Main 3rd Sep I 2020


Solution


A

B

12

4

D

C

10

-8

A

B

D

C

D

JEE Main 3rd Sep I 2020

Q1.


Q2. Let . If B = A + A4 then det (B) :

A

B

is zero

is one

D

C

lies in (1, 2)

lies in (2, 3)

A

B

D

C

D

JEE Main 6th Sep II 2020


Solution


Solution


A

B

is zero

is one

D

C

lies in (1, 2)

lies in (2, 3)

A

B

D

C

D


Q2. Let . If B = A + A4 then det (B) :


Q3. Let S be the set of all 𝜆∊R for which the system of linear equations 2x - y +2z = 2, x - 2y + 𝜆z = -4, x + 𝜆y +z =4, has no solution. Then the set S

Is a singleton

Contains more than two elements

Is an empty set

Contains exactly two elements

A

B

D

C

JEE Main 2nd Sep I 2020


Solution


Q3. Let S be the set of all 𝜆∊R for which the system of linear equations 2x - y +2z = 2, x - 2y + 𝜆z = -4, x + 𝜆y +z =4, has no solution. Then the set S

Is a singleton

Contains more than two elements

Is an empty set

Contains exactly two elements

A

B

D

C

JEE Main 2nd Sep I 2020



Q4. If the system of equations x - 2y + 3z = 9, 2x + y + z = b,

x - 7y + az = 24, has infinitely many solutions, then a - b is equal to:

JEE Main 4th Sep I 2020


Solution


Q5. If the system of equations has infinitely many

solutions then

A

B

D

C

λ + 2μ = 14

2λ - μ = 5

2λ + μ = 14

λ - 2μ = -5



Solution


Q5. If the system of equations has infinitely many

solutions then

A

B

D

C

λ + 2μ = 14

2λ - μ = 5

2λ + μ = 14

λ - 2μ = -5



Q6. Let λ ∈ R. The system of linear equations 2x1 - 4x2 + λx3 = 1, x1 - 6x2 + x3 = 2λx1 - 10x2 + 4x3 = 3 is inconsistent for:

A

B

D

C

A

B

D

C

D

Every value of λ

Exactly two values of λ

Exactly one negative value of λ

Exactly one positive value of λ



Solution


Q6. Let λ ∈ R. The system of linear equations 2x1 - 4x2 + λx3 = 1, x1 - 6x2 + x3 = 2λx1 - 10x2 + 4x3 = 3 is inconsistent for:

A

B

D

C

A

B

D

C

D

Every value of λ

Exactly two values of λ

Exactly one negative value of λ

Exactly one positive value of λ




Q7. The sum of distinct values of λ for which the system of equations: (λ – 1)x + (3λ + 1)y + 2λz = 0(λ – 1)x + (4λ – 2)y + (λ + 3)z = 0 2x + (3λ + 1)y + 3(λ – 1)z = 0, Has non-zero solutions, is..........



Solution


Q8 . Let A be a 3 x 3 matrix such that adj and

B = adj(adj A). If |A| = λ and |(B-1 )T |= μ,

then the ordered pair, (| λ|,μ) is equal to

A

B

D

C

A

B

D

C

D

JEE Main 3rd Sep II 2020


Solution


Q8 . Let A be a 3 x 3 matrix such that adj and

B = adj(adj A). If |A| = λ and |(B-1 )T = μ, then the ordered pair, (| λ|,μ)

is equal to

A

B

D

C

A

B

D

C

D

JEE Main 3rd Sep II 2020


Q9. Let a, b, c ∈ R be all non-zero satisfy a3 + b3 + c3 = 2. If the matrix

satisfies ATA = I, then a value of abc can be :

A

B

D

C 3

A

B

D

C

D

JEE Main 2nd Sep II 2020


Solution


Q9. Let a, b, c ∈ R be all non-zero satisfy a3 + b3 + c3 = 2. If the matrix

stratifies ATA = I, then a value of abc can be :

A

B

D

C 3

A

B

D

C

D

JEE Main 2nd Sep II 2020


Q10. If a + x = b + y + 1= c + z, where a, b, c, x, y, z are non-zero distinct

real numbers, then is equal to :

A

B

D

C

A

B

D

C

D

y (a - b)

0

y (a - c)

y (b - a)



Solution


Solution


Q10. If a + x = b + y + 1= c + z, where a, b, c, x, y, z are non-zero distinct

real numbers, then is equal to :

A

B

D

C

A

B

D

C

D

y (a - b)

0

y (a - c)

y (b - a)



JEE MAIN 2021 CRASH COURSE Course Overview

1. Cover entire JEE Main Syllabus (as per the new pattern) with India’s Best Teachers in 90 Sessions

2. Solve unlimited doubts with Doubt experts on our Doubt App from 8 AM to 11 PM

3. 10 full and 10 Part syllabus test to make you exam ready

4. Amazing tricks & tips to crack JEE Main 2021 Questions in power-packed 90 Min sessions

5. Learn on a 2-way Interactive Platform where the teacher is always with you

JEE MAIN 2021 CRASH COURSELightning Deal: ₹10000 - ₹8000/-

Batch Starts From : 16th Nov 2020

*Crash Course Link Available in Description

Apply Coupon Code: NAGCC





Thank You

Documents

JEE MAIN 2021: MATRICES & DETERMINANTS in ONE SHOT