If A = , verify that A2 – 4A – I = O, and hence find A-1.
Given:
To verify: A2 – 4A – I = 0
Firstly, we find the A2
Taking LHS of the given equation .i.e.
A2 – 4A – I
= 0
= RHS
∴ LHS = RHS
Hence verified
Now, we have to find A-1
Finding A-1 using given equation
A2 – 4A – I = O
Post multiplying by A-1 both sides, we get
(A2 – 4A – I)A-1 = OA-1
⇒ A2.A-1 – 4A.A-1 – I.A-1 = O [OA-1 = O]
⇒ A.(AA-1) – 4I – A-1 = O [AA-1 = I]
⇒ A(I) – 4I – A-1 = O
⇒ A – 4I – A-1 = O
⇒ A – 4I – O = A-1
⇒ A – 4I = A-1
Ans.