# matrices

## 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 ( 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