# 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 = , then show that A^{2} = –I

∴ A^{2} = –I

**QUESTION 2**

If A = , and A^{2} = 0, then find the value of k.

A^{2} = 0

8 + 4k = 0, – 2 – k = 0 and –4 + k^{2} = 0

4k = –8; k = –2; k^{2} = 4

k = –2; k = –2; k = ± 2

∴ k =– 2

**QUESTION 3**

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

**QUESTION 5**

Find the additive inverse of A, If A =

Additive inverse of A = – A

**QUESTION 6**

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

⟹ 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 a_{ij} =

**Sol:**

Let A= _{ }

a_{11} = 1

a_{22} = 2

a_{31} = 0

**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 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 A^{T} = A, then A is called Symmetric Matrix

Skew Symmetric Matrix: Let A be any square matrix if A^{T} = –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

⟹ A^{T} = A

⟹ x = 6

**QUESTION 11**

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

**Sol:** Given A = is a skew-symmetric matrix

⟹ A^{T} = – A

⟹ x = –x

x+ x = 0 ⟹ 2x = 0

⟹ x = 0

**QUESTION 12**

If A = and B = , then find (A B^{T})^{ T}

**QUESTION 13**

If A = and B = , then find A + B^{T}

**QUESTION 14**

If A = , then show that AA^{T} = A^{T}A = I

∴ AA^{T} = A^{T}A = I

**QUESTION 15**

Find the minor of – 1 and 3 in the matrix

**Sol:** Given Matrix is

**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+ω^{2} = 0)

R_{1} → R_{1} + R_{2} + R_{3}

**QUESTION 18**

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

Det A = 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)

(ii)

**QUESTION 20**

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

Det A = abc ≠ 0

Adj A = (Cofactor matrix of A)^{ T}

**QUESTION 20**

Find the rank of the following matrices.

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

= – 2– 2– 1

= – 5 ≠ 0

∴ Rank of A = 3

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

= – 1– 4 + 3

= – 0

Det B = – 4 + 6 = 2 ≠ 0

∴ Rank of A = 2

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

∴ Rank of A = 2

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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