matrices

ts inter matrices 4 marks important questions 2024

ts inter || matrices 4 marks important questions 2024



ts inter || matrices 4 marks important questions 2024

Matrices  

Here are some important questions related to matrices that could be worth 4 marks each. Keep in mind that the specific marking scheme may vary based on the curriculum and exam format. These questions cover various aspects of matrices, including operations, properties, and applications:

 

Matrices 4m imp qs link - 1

Properties of Triangles 4m imp qs link - 1

Maths IA Two Marks Questions & Solutions  link 

Maths IB Two Marks Questions & Solutions  link

matrices 4m - 1 -1       matrices 4m - 2

 

Addition of Vectors 4m imp qs link - 1

Trigonometric Equations 4m imp qs link - 1

Maths – IA Concept link

Maths – IB Concept link

 

matrices 4m - 2

 

Inverse Trigonometric Functions 4m imp qs link - 1

Trigonometric Ratios Up To Transformations 4m imp qs link - 1

Maths – IIB Concept link Maths – IIA Concept link

 

matrices 4m - 5

 

Properties of Triangles 4m imp qs link - 1


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Matrices vsaqs questions and solutions

Matrices ( Qns & Solutions) || V.S.A.Q’S||

Matrices ( Qns & Solutions) || V.S.A.Q’S||

Matrices V.S.A.Q’s: This note is designed by the ‘Basics in Maths’ team. These notes to do help intermediate First-year Maths students.

Inter Maths – 1A two marks questions and solutions are very useful in IPE examinations.

These notes cover all the topics covered in the intermediate First-year Maths syllabus and include plenty of solutions to help you solve all the major types of Math problems asked in the

IPE examinations.  


Matrices

QUESTION 1

If A = Matrices 1, then show that A2 = –I

Sol: Given A = Matrices 1

Matrices 2

  ∴  A2 = –I

QUESTION 2

If A = Matrices 3, and A2 = 0, then find the value of k.

Sol: Given A = Matrices 3 and

 A2 = 0

⟹ A. A =0 ⟹  Matrices 4   = 0

Matrices 5 = 0

     8 + 4k = 0, – 2 – k = 0 and –4 + k2 = 0

    4k = –8; k = –2; k2 = 4

       k = –2; k = –2; k = ± 2

   ∴ k =– 2

QUESTION 3

Find the Trace of A, If A = Matrices 6

Sol: Given A =Matrices 6

       Trace of A = 1 – 1 + 1 = 1

QUESTION 4

If A =Matrices 12 , B = Matrices 13and 2X + A = B, then find X.

Sol: Given A =Matrices 12 , B = Matrices 13 and 2X + A = B

        2X = B – A

        2X =Matrices 13  – Matrices 12

              = Matrices 8

              =Matrices 9

           X =   Matrices 10 Matrices 9

         ∴ X =   Matrices 11

QUESTION 5

Find the additive inverse of A, If A =Matrices 14

Sol: Given A =Matrices 14

       Additive inverse of A = – A

    = –Matrices 14

   =Matrices 15

QUESTION 6

If Matrices 16, then find the values of x, y, z and a.

Sol: Given Matrices 16

 ⟹ x- 1 = 1 – x ; y – 5 =  – y ; z = 2 ; 1 + a = 1

 ⟹ x + x = 1 + 1; y + y = 5; z = 2; a =1– 1

  ⟹ 2x = 1; 2y = 5; z = 2; a = 0

∴ x = ½ ; y = 5/2; z = 2; a = 0

QUESTION 7

Construct 3 × 2 matrix whose elements are defined by aij =Matrices 17

Sol:

Let A= Matrices 18

a11 = Matrices 19

a11 = 1

a12 = Matrices 20

a12 =Matrices 22

a21 = Matrices 23

a21 =Matrices 24

a22 = Matrices 25

a22 = 2

a31 = Matrices 26

a31 = 0

a32 = Matrices 27

a32 =Matrices 28

 ∴ A =Matrices 29

QUESTION 8

If A = Matrices 30 and B = Matrices 31, do AB and BA exist? If they exist, find them. BA and AB commutative with respect to multiplication.

Sol: Given Matrices are A = Matrices 30 B =Matrices 31

       Order of A = 2 × 3 and Order of B = 3 × 2

AB and BA exist

 

 AB =   Matrices 30Matrices 31

    Matrices 32

BA =     Matrices 31Matrices 30

Matrices 33

 AB and  BA are not Commutative under Multiplication 

QUESTION 9

Define Symmetric and Skew Symmetric Matrices

Sol:

Symmetric Matrix: Let A be any square matrix, if AT = A, then A is called Symmetric Matrix

Skew Symmetric Matrix: Let A be any square matrix if AT = –A, then A is called Skew Symmetric Matrix

QUESTION 10

If A =Matrices 34 is a symmetric matrix, then find x.

Sol: Given, A = Matrices 34 is a symmetric matrix

       ⟹ AT = A

          Matrices 35     

          ⟹ x = 6

QUESTION 11

If A =Matrices 36 is a skew-symmetric matrix, then find x

Sol: Given A = Matrices 36is a skew-symmetric matrix

       ⟹ AT = – A

     

        ⟹ x = –x

        x+ x = 0 ⟹ 2x = 0

     ⟹ x = 0

QUESTION 12

If A =Matrices 38 and B = Matrices 39, then find (A BT) T

Sol: Given A = Matrices 38   B =Matrices 39

   BT = Matrices 40   

   (A BT) =  Matrices 38   Matrices 40

                = Matrices 41

(A BT) T = Matrices 42

QUESTION 13

If A =Matrices 43 and B =Matrices 44 , then find A + BT

Sol: Given A =Matrices 43  and B =Matrices 44

 BT =Matrices 45

A + BT = Matrices 43 + Matrices 45

 Matrices 47            

QUESTION 14

If A = Matrices 48, then show that AAT = ATA = I

Sol: Given A =Matrices 48

  AT =Matrices 49

AAT =

= Matrices 51

 = Matrices 52 

ATA =Matrices 49Matrices 48

        =

       =Matrices 52

∴ AAT = ATA = I

QUESTION 15

Find the minor of – 1 and 3 in the matrixMatrices 54

Sol: Given Matrix is

       minor of – 1 = Matrices 55 = 0 + 15 = 15

     minor of 3 = Matrices 56 = – 4 + 0 = – 4

QUESTION 16

Find the cofactors 0f 2, – 5 in the matrixMatrices 57

Sol: Given matrix is

 Cofactor of 2 = (–1)2 + 2 Matrices 58= –3 + 20 = 17

  Cofactor of – 5 = (–1)3 + 2  Matrices 59= –1(2 – 5) = –1(–3) = 3

QUESTION 17

If ω is a complex cube root of unity, then show that Matrices 62= 0(where 1 + ω+ω2 = 0)

Given matrix is    Matrices 62

   R1 → R1 + R2 + R3

   Matrices 73 Matrices 74

Matrices 74 = 0 (∵ 1 + ω+ω2 = 0)

QUESTION 18

If A = Matrices 63and det A = 45, then find x.

Sol: Given A = Matrices 63

Det A = 45

Matrices 64= 45

   ⟹ 1(3x + 24) – 0 (2x – 20) + 0 (– 12 – 15) = 45

 ⟹ 3x + 24 = 45

        3x = 45 – 24

        3x = 21

         x = 7

QUESTION 19

Find the adjoint and inverse of the following matrices

(i)

A =Matrices 65

Adj A =Matrices 66

A-1 = Matrices 67

       =Matrices 68

   ∴ A-1 =Matrices 69

(ii)

A =Matrices 70

Adj A =Matrices 71

A-1 =Matrices 72

 ∴ A-1 = Matrices 71   

QUESTION 20

Find the inverse of Matrices 75 (abc ≠ 0)

Sol: Let A =Matrices 75

        Det A = a (bc – 0) – 0(0 – 0) + 0(0 – 0)

        Det A = abc ≠ 0

Cofactor matrix of A =Matrices 76

Adj A = (Cofactor matrix of A) T

           =Matrices 76

A-1 =Matrices 77

  A-1 = Matrices 85 Matrices 76

 ∴ A-1Matrices 78

QUESTION 20

Find the rank of the following matrices.

(i) Matrices 79

Let A =Matrices 79

Det A = 1 (0 – 2) – 2(1 – 0) + 1(– 1 – 0)

           = – 2– 2– 1

           = – 5 ≠ 0

∴ Rank of A = 3

(ii)Matrices 80

Let A =Matrices 80

Det A = – 1 (24 – 25) + 2(18 – 20) + – 3(15 – 16)

           = – 1– 4 + 3

           = – 0

Sub matrix of A = Matrices 81

      Let B =Matrices 81

       Det B = – 4 + 6 = 2 ≠ 0

          ∴ Rank of A = 2

(iii)Matrices 82

Let A =Matrices 82

Sub matrix of A = Matrices 86

Det of Sub matrix of A = – 1 – 0 = – 1 ≠ 0

      ∴ Rank of A = 2

(iv)Matrices 83

Let A =Matrices 83

Sub matrix of A =Matrices 84

Det of Sub matrix of A =1 (1 – 0) – 0(0 – 0) + 0(0 – 0)

                                           = 1≠ 0

      ∴ Rank of A = 3


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