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.