We are given that,

A² = pA

We need to find the value of p.

First, let us find A².

We know that, A² = A.A

In multiplication of matrices A and A, such that A² = Z(say):

For the calculation of z₁₁: Dot multiply the 1^st row of first matrix and the 1^st column of second matrix and then sum up.

(2 -2)(2 -2) = 2 × 2 + (-2) × (-2)

⇒ (2 -2)(2 -2) = 4 + 4

⇒ (2 -2)(2 -2) = 8

So,

For the calculation of z₁₂: Dot multiply the 1^st row of first matrix and the 2^nd column of second matrix and then sum up.

(2 -2)(-2 2) = 2 × -2 + (-2) × 2

⇒ (2 -2)(-2 2) = -4 – 4

⇒ (2 -2)(-2 2) = -8

So,

Similarly,

So,

…(i)

Now, let us find pA.

Multiply p by matrix A,

…(ii)

Substituting value of A² and pA from (i) and (ii) in

A² = pA

We know by the property of matrices,

This implies,

a₁₁ = b₁₁, a₁₂ = b₁₂, a₂₁ = b₂₁ and a₂₂ = b₂₂

So,

2p = 8

-2p = -8

-2p = -8

2p = 8

Take equation,

2p = 8

⇒ p = 4

Thus, the value of p = 4.