Given:

To find: value of λ

Given equation: A² = λA – 2I

Firstly, we find the A²

Putting the values in given equation

A² = λA – 2I

On Comparing, we get

3λ – 2 = 1 …(i)

-2λ = -2 …(ii)

4λ = 4 …(iii)

-2λ – 2 = -4 …(iv)

Solving eq. (iii), we get

4λ = 4

⇒ λ = 1

Hence, the value of λ = 1

So, the given equation become A² = A – 2I

Now, we have to find A^-1

Finding A^-1 using given equation

A² = A – 2I

Post multiplying by A^-1 both sides, we get

(A²)A^-1 = (A – 2I) A^-1

⇒ A².A^-1 = AA^-1 – 2IA^-1

⇒ A.(AA^-1) = I – 2A^-1 [AA^-1 = I]

⇒ A(I) = I – 2A^-1

⇒ A + 2A^-1 = I

⇒ 2A^-1 = – A + I

Ans. λ= 1, A^-1 = .