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

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This note is designed by the ‘Basics in Maths’ team. These notes to do help the 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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