The number of elements in the set A = a b 0 d :   a ,   b ,   d ∈ - 1 ,   0 ,   1 and ( I - A ) 3 = I - A 3 , where I is 2 × 2 identity matrix, is .

Step-by-step solution

Given, A = a b 0 d   : a , b , d ∈ - 1 , 0 , 1 I - A 3 = I - A 3 I - A 3 - 3 A + 3 A 2 = I - A 3 3 A 2 - 3 A = 0 3 A ( A - I ) = 0 A 2 = A . . . . i A 2 = A . A = a b 0 d a b 0 d A 2 = a 2 a b + b d 0 d 2 From i a 2 a b + b d 0 d 2 = a b 0 d Comparing on both sides a 2 = a a a - 1 = 0 a = 0 ,   1 d 2 = d d d - 1 = 0 d = 0 ,   1 b ( a + d ) = b Case I : b = 0 ⇒ ( a ,   d ) ≡ ( 0 ,   1 ) ,   ( 0 ,   0 ) ,   ( 1 ,   1 ) → 4 ways Case II   : a +