Answer :

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 1st row of first matrix and the 1st 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 1st row of first matrix and the 2nd 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.


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