Here,

Now, we have to find adj A and for that we have to find co-factors:

a₁₁ (co – factor of cos α) = (-1)¹⁺¹(cos α)= (-1)²(cos α) = cos α

a₁₂ (co – factor of sin α) = (-1)¹⁺²(sin α) = (-1)³(sin α) = -sin α

a₂₁ (co – factor of sin α) = (-1)²⁺¹(sin α) = (-1)³(sin α) = -sin α

a₂₂ (co – factor of cos α) = (-1)²⁺²(cos α)= (-1)⁴(cos α) = cos α

Now, adj A = Transpose of co-factor Matrix

Calculating A (adj A)

= (cos² α – sin² α) I

Calculating (adj A)A

= (cos² α – sin² α) I

Calculating |A|.I

= [cos α × cos α – (sin α) × (sin α)]I

= [cos² α – sin² α] I

Thus, A(adj A) = (adj A)A = |A|I = I

⇒ A(adj A) = (adj A)A = |A|I

Hence Proved

Ans.