Chapter 2 Matrices Ex 2.2

Question 1.
Find the co-factors of the elements of the following matrices

Here, a₁₁ = -11, M₁₁ = 4
∴ A₁₁ = (-1)¹⁺¹(4) = 4
a₁₂ = 2, M₁₂ = -3
∴ A₁₂ = (-1)¹⁺²(- 3) = 3
a₂₁ = – 3, M₂₁ = -2
∴ A₂₁ = (- 1)²⁺¹(2) = -2
a₂₂ = 4, M₂₂ = -1
∴ A₂₂ = (-1)²⁺²(-1) = -1.

The co-factor of a_ij is given by A_ij = (-1)^i+jM_ij

Question 2.
Find the matrix of co-factors for the following matrices

Here, a₁₁ = 1, M₁₁ = -1
∴ A₁₁ = (-1)¹⁺¹(-1) = -1
a₁₂ = 3, M₁₂ = 4
∴ A₁₂ = (-1)¹⁺²(4) = -4
a₂₁ = 4, M₂₁ = 3
∴ A₂₁ = (-1)²⁺¹(3) = -3
a₂₂ = -1, M₂₂ = 1
∴ A₂₂ = (-1)²⁺¹(1) = 1




A₁₁ = -14, A₁₂ = -10, A₁₃ = -6,
A₂₁ = 6, A₂₂ = -5, A₂₃ = -3,
A₃₁ = -2, A₃₂ = -7, A₃₃ = 1.
∴ the co-factor matrix

Question 3.
Find the adjoint of the following matrices.

Here, a₁₁ = 2, M₁₁= 5
∴ A₁₁ = (-1)¹⁺¹(5) = 5
a₁₂ = -3, M₁₂ = 3
∴ A₁₂ = (-1)¹⁺²(3) = -3
a₂₁ = 3, M₂₁ = -3
∴ A A₂₁ = (-1)²⁺¹(-3) = 3
a₂₂ = 5, M₂₂ = 2
∴ A₂₂ = (-1)²⁺¹ = 2

Solution:


A₁₁ = -3, A₁₂ = -12, A₁₃ = 6,
A₂₁ = -1, A₂₂ = 3, A₂₃ = 2,
A₃₁ = -11, A₃₂ = -9, A₃₃ = 1





From (1), (2) and (3), we get,
A(adj A) = (adj A)A = |A|∙I.
Note: This relation is valid for any non-singular matrix A.

Question 5.
Find the inverse of the following matrices by the adjoint method

∴ A^-1 exists.
First we have to find the co-factor matrix
= [A_ij]_2×2, where A_ij = (-1)^i+jM_ij
Now, A₁₁ = (-1)¹⁺¹M₁₁ = 2
A₁₂ = (-1)¹⁺²M₁₂ = -(-3) = 3
A₂₁ = (-1)²⁺¹M₂₁ = -5
A₂₂ = (-1)²⁺²M₂₂ = -1
Hence, the co-factor matrix

|A| =  = 6 + 8 = 14 ≠ 0
∴ A^-1 exist
First we have to find the co-factor matrix
= [A_ij] _2×2 where A_ij = (-1)^i+jM_ij
Now, A₁₁ = (-1)¹⁺¹M₁₁ = 3
A₁₂ = (-1)¹⁺²M = -4
A₂₁ = (-2)²⁺¹M₂₁ = (-2) = 2
A₂₂ = (-1)²⁺²M₂₂ = 2
Hence the co-factor matrix

= 1(10 – 0) – 0 + 0
= 1(10) – 0 + 0
= 10 ≠ 0
∴ A^-1 exists.
First we have to find the co-factor matrix

Question 6.
Find the inverse of the following matrices