## 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 = , then show that A2 = –I

Sol: Given A =

∴  A2 = –I

#### QUESTION 2

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

Sol: Given A =  and

A2 = 0

⟹ A. A =0 ⟹     = 0

= 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 =

Sol: Given A =

Trace of A = 1 – 1 + 1 = 1

##### QUESTION 4

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

Sol: Given A = , B =  and 2X + A = B

2X = B – A

2X =  –

=

=

X =

∴ X =

#### QUESTION 5

Find the additive inverse of A, If A =

Sol: Given A =

Additive inverse of A = – A

= –

=

##### QUESTION 6

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

Sol: Given

⟹ 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 =

Sol:

Let A=

a11 =

a11 = 1

a12 =

a12 =

a21 =

a21 =

a22 =

a22 = 2

a31 =

a31 = 0

a32 =

a32 =

∴ A =

#### QUESTION 8

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

Sol: Given Matrices are A =  B =

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

AB and BA exist

AB =

BA =

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 = is a symmetric matrix, then find x.

Sol: Given, A =  is a symmetric matrix

⟹ AT = A

⟹ x = 6

#### QUESTION 11

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

Sol: Given A = is a skew-symmetric matrix

⟹ AT = – A

⟹ x = –x

x+ x = 0 ⟹ 2x = 0

⟹ x = 0

##### QUESTION 12

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

Sol: Given A =    B =

BT =

(A BT) =

=

(A BT) T =

#### QUESTION 13

If A = and B = , then find A + BT

Sol: Given A =  and B =

BT =

A + BT = +

##### QUESTION 14

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

Sol: Given A =

AT =

AAT =

=

=

ATA =

=

=

∴ AAT = ATA = I

###### QUESTION 15

Find the minor of – 1 and 3 in the matrix

Sol: Given Matrix is

minor of – 1 =  = 0 + 15 = 15

minor of 3 =  = – 4 + 0 = – 4

#### QUESTION 16

Find the cofactors 0f 2, – 5 in the matrix

Sol: Given matrix is

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

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

#### QUESTION 17

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

Given matrix is

R1 → R1 + R2 + R3

= 0 (∵ 1 + ω+ω2 = 0)

##### QUESTION 18

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

Sol: Given A =

Det A = 45

= 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 =

A-1 =

=

∴ A-1 =

(ii)

A =

A-1 =

∴ A-1 =

##### QUESTION 20

Find the inverse of (abc ≠ 0)

Sol: Let A =

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

Det A = abc ≠ 0

Cofactor matrix of A =

Adj A = (Cofactor matrix of A) T

=

A-1 =

A-1 =

∴ A-1

#### QUESTION 20

Find the rank of the following matrices.

(i)

Let A =

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

= – 2– 2– 1

= – 5 ≠ 0

∴ Rank of A = 3

(ii)

Let A =

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

= – 1– 4 + 3

= – 0

Sub matrix of A =

Let B =

Det B = – 4 + 6 = 2 ≠ 0

∴ Rank of A = 2

(iii)

Let A =

Sub matrix of A =

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

∴ Rank of A = 2

(iv)

Let A =

Sub matrix of A =

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

= 1≠ 0

∴ Rank of A = 3