Question - 3 : -
Let A be a matrix oforder 3 × 4. If R₁ denotes the first row of A and C₂ denotesits second column, then determine the orders of matrices R₁ andC₂.

Answer - 3 : -

Given A be a matrix of order 3 × 4.

So, A = [a_{i j}] _3×4

R₁ = first row of A = [a₁₁, a₁₂,a₁₃, a₁₄]

So, order of matrix R₁ = 1 × 4

C₂ = second column of

Therefore order of C₂ = 3 × 1

Question - 4 : -
Construct a 2 ×3matrix A = [a_{j j}] whose elements a_{j j} are given by:
(i) a_{i j} =i × j
(ii) a_ij= 2i – j
(iii) a_ij= i + j
(iv) a_{i j} =(i + j)²/2

Answer - 4 : -

(i) Given a_{i j} = i × j

Let A = [a_{i j}]_{2 × 3}

So, the elements in a 2 × 3 matrix are

[a₁₁, a₁₂,a₁₃, a₂₁, a₂₂, a₂₃]

a₁₁ = 1 × 1 = 1

a₁₂ = 1 × 2 = 2

a₁₃ = 1 × 3 = 3

a₂₁ = 2 × 1 = 2

a₂₂ = 2 × 2 = 4

a₂₃ = 2 × 3 = 6

Substituting these values in matrix A we get,

(ii) Given a_{i j}= 2i – j

Let A = [a_{i j}]_2×3

So, the elements in a 2 × 3 matrix are

a₁₁, a₁₂, a₁₃, a₂₁,a₂₂, a₂₃

a₁₁ = 2 × 1 – 1 = 2 – 1 = 1

a₁₂ = 2 × 1 – 2 = 2 – 2 = 0

a₁₃ = 2 × 1 – 3 = 2 – 3 = – 1

a₂₁ = 2 × 2 – 1 = 4 – 1 = 3

a₂₂ = 2 × 2 – 2 = 4 – 2 = 2

a₂₃ = 2 × 2 – 3 = 4 – 3 = 1

Substituting these values in matrix A we get,

(iii) Given a_{i j}= i + j

Let A = [a _{i j}] _2×3

So, the elements in a 2 × 3 matrix are

a₁₁, a₁₂, a₁₃, a₂₁,a₂₂, a₂₃

a₁₁ = 1 + 1 = 2

a₁₂ = 1 + 2 = 3

a₁₃ = 1 + 3 = 4

a₂₁ = 2 + 1 = 3

a₂₂ = 2 + 2 = 4

a₂₃ = 2 + 3 = 5

Substituting these values in matrix A we get,

(iv) Given a_{i j} = (i + j)²/2

Let A = [a_{i j}]_2×3

So, the elements in a 2 × 3 matrix are

a₁₁, a₁₂, a₁₃, a₂₁,a₂₂, a₂₃

Let A = [a_{i j}]_2×3

So, the elements in a 2 × 3 matrix are

a₁₁, a₁₂, a₁₃, a₂₁,a₂₂, a₂₃

a₁₁ =
a₁₂ =

a₁₃ =
a₂₁ =
a₂₂ =

a₂₃ =

Substituting these values in matrix A we get,

Question - 5 : -
Construct a 2 × 2matrix A = [a_{i j}] whose elements a_{i j} are given by:
(i) (i + j)²/2
(ii) a_{i j} =(i – j)²/2
(iii) a_{i j} =(i – 2j)²/2
(iv) a_{i j} =(2i + j)²/2
(v) a_{i j} =|2i – 3j|/2
(vi) a_{i j} =|-3i + j|/2
(vii) a_{i j} =e^2ix sin x j

Answer - 5 : -

(i) Given (i + j)²/2

Let A = [a_{i j}]_2×2

So, the elements in a 2 × 2 matrix are

a₁₁, a₁₂, a₂₁, a₂₂

a₁₁ =

a₁₂ =

a₂₁ =

a₂₂ =

Substituting these values in matrix A we get,

(ii) Given a_{i j} = (i – j)²/2

Let A = [a_{i j}]_2×2

So, the elements in a 2 × 2 matrix are

a₁₁, a₁₂, a₂₁, a₂₂

a₁₁ =

a₁₂ =

a₂₁ =

a₂₂ =

Substituting these values in matrix A we get,

(iii) Given a_{i j} = (i – 2j)²/2

Let A = [a_{i j}]_2×2

So, the elements in a 2 × 2 matrix are

a₁₁, a₁₂, a₂₁, a₂₂

a₁₁ =

a₁₂ =

a₂₁ =

a₂₂ =

Substituting these values in matrix A we get,

(iv) Given a_{i j} = (2i + j)²/2

Let A = [a_{i j}]_2×2

So, the elements in a 2 × 2 matrix are

a₁₁, a₁₂, a₂₁, a₂₂

a₁₁ =

a₁₂ =

a₂₁ =

a₂₂ =

Substituting these values in matrix A we get,

(v) Given a_{i j} = |2i – 3j|/2

Let A = [a_{i j}]_2×2

So, the elements in a 2×2 matrix are

a₁₁, a₁₂, a₂₁, a₂₂

a₁₁ =

a₁₂ =

a₂₁ =

a₂₂ =

Substituting these values in matrix A we get,

(vi) Given a_{i j} = |-3i + j|/2

Let A = [a_{i j}]_2×2

So, the elements in a 2 × 2 matrix are

a₁₁, a₁₂, a₂₁, a₂₂

a₁₁ =

a₁₂ =

a₂₁ =

a₂₂ =

Substituting these values in matrix A we get,

(vii) Given a_{i j} = e^2ix sin x j

Let A = [a_{i j}]_2×2

So, the elements in a 2 × 2 matrix are

a₁₁, a₁₂, a₂₁, a₂₂,

a₁₁ =

a₁₂ =

a₂₁ =

a₂₂ =

Substituting these values in matrix A we get,

a₃₃ =

a₃₄ =

Substituting these values in matrix A we get,

A =

Multiplying by negative sign we get,

Question - 6 : -
Construct a 4 × 3matrix A = [a_{i j}] whose elements a_{i j} are given by:
(i) a_{i j} =2i + i/j
(ii) a_{i j} =(i – j)/ (i + j)
(iii) a_{i j} =i

Answer - 6 : -

(i) Given a_{i j} = 2i + i/j

Let A = [a_{i j}]_4×3

So, the elements in a 4 × 3 matrix are

a₁₁, a₁₂, a₁₃, a₂₁,a₂₂, a₂₃, a_31,a_32,a_33,a_41, a_42, a₄₃

A =

a₁₁ =

a₁₂ =

a₁₃ =

a₂₁ =

a₂₂ =

a₂₃ =

a₃₁ =

a₃₂ =

a₃₃ =

a₄₁ =

a₄₂ =

a₄₃ =

Substituting these values in matrix A we get,

A =

(ii) Given a_{i j} = (i – j)/ (i + j)

Let A = [a_{i j}]_4×3

So, the elements in a 4 × 3 matrix are

a₁₁, a₁₂, a₁₃, a₂₁,a₂₂, a₂₃, a_31,a_32,a_33,a_41, a_42, a₄₃

A =

a₁₁ =

a₁₂ =

a₁₃ =

a₂₁ =

a₂₂ =

a₂₃ =

a₃₁ =

a₃₂ =

a₃₃ =

a₄₁ =

a₄₂ =

a₄₃ =

Substituting these values in matrix A we get,

A =

(iii) Given a_{i j} = i

Let A = [a_{i j}]_4×3

So, the elements in a 4 × 3 matrix are

a₁₁, a₁₂, a₁₃, a₂₁,a₂₂, a₂₃, a_31,a_32,a_33,a_41, a_42, a₄₃

A =

a₁₁ = 1

a₁₂ = 1

a₁₃ = 1

a₂₁ = 2

a₂₂ = 2

a₂₃ = 2

a₃₁ = 3

a₃₂ = 3

a₃₃ = 3

a₄₁ = 4

a₄₂ = 4

a₄₃ = 4

Substituting these values in matrix A we get,

A =

Question - 7 : - Find x, y, a and b if

Answer - 7 : -

Given

Given that two matrices are equal.

We know that if two matrices are equal then the elements of eachmatrices are also equal.

Therefore by equating them we get,

3x + 4y = 2 …… (1)

x – 2y = 4 …… (2)

a + b = 5 …… (3)

2a – b = – 5 …… (4)

Multiplying equation (2) by 2 and adding to equation (1), we get

3x + 4y + 2x – 4y = 2 + 8

⇒ 5x= 10

⇒ x= 2

Now, substituting the value of x in equation (1)

3 × 2 + 4y = 2

⇒ 6+ 4y = 2

⇒ 4y= 2 – 6

⇒ 4y= – 4

⇒ y= – 1

Now by adding equation (3) and (4)

a + b + 2a – b = 5 + (– 5)

⇒ 3a= 5 – 5 = 0

⇒ a= 0

Now, again by substituting the value of a in equation (3), weget

0 + b = 5

⇒ b= 5

∴ a= 0, b = 5, x = 2 and y = – 1

Question - 8 : -
Find x, y, a and bif

Answer - 8 : -

We know that if two matrices are equal then the elements of eachmatrices are also equal.

Given that two matrices are equal.

Therefore by equating them we get,

2a + b = 4 …… (1)

And a – 2b = – 3 …… (2)

And 5c – d = 11 …… (3)

4c + 3d = 24 …… (4)

Multiplying equation (1) by 2 and adding to equation (2)

4a + 2b + a – 2b = 8 – 3

⇒ 5a= 5

⇒ a= 1

Now, substituting the value of a in equation (1)

2 × 1 + b = 4

⇒ 2+ b = 4

⇒ b= 4 – 2

⇒ b= 2

Multiplying equation (3) by 3 and adding to equation (4)

15c – 3d + 4c + 3d = 33 + 24

⇒ 19c= 57

⇒ c= 3

Now, substituting the value of c in equation (4)

4 × 3 + 3d = 24

⇒ 12+ 3d = 24

⇒ 3d= 24 – 12

⇒ 3d= 12

⇒ d= 4

∴ a= 1, b = 2, c = 3 and d = 4

Question - 9 : -
Find the values ofa, b, c and d from the following equations:

Answer - 9 : -

Given

We know that if two matrices are equal then the elements of eachmatrices are also equal.

Given that two matrices are equal.

Therefore by equating them we get,

2a + b = 4 …… (1)

And a – 2b = – 3 …… (2)

And 5c – d = 11 …… (3)

4c + 3d = 24 …… (4)

Multiplying equation (1) by 2 and adding to equation (2)

4a + 2b + a – 2b = 8 – 3

⇒ 5a= 5

⇒ a= 1

Now, substituting the value of a in equation (1)

2 × 1 + b = 4

⇒ 2+ b = 4

⇒ b= 4 – 2

⇒ b= 2

Multiplying equation (3) by 3 and adding to equation (4)

15c – 3d + 4c + 3d = 33 + 24

⇒ 19c= 57

⇒ c= 3

Now, substituting the value of c in equation (4)

4 × 3 + 3d = 24

⇒ 12+ 3d = 24

⇒ 3d= 24 – 12

⇒ 3d= 12

⇒ d= 4

∴ a= 1, b = 2, c = 3 and d = 4

Question - 10 : -

Answer - 10 : -

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