Let X = 0 1 0 0 0 1 0 0 0 , Y = α l + β X + γ X 2 and Z = α 2 I - α β X + β 2 - α γ X 2 , α , β , γ ∈ ℝ . If Y - 1 = 1 5 - 2 5 1 5 0 1 5 - 2 5 0 0 1 5 , then α - β + γ 2 is equal to ______.

Step-by-step solution

Given, X = 0 1 0 0 0 1 0 0 0 , so X 2 = 0 0 1 0 0 0 0 0 0 Now, finding Y = α l + β X + γ X 2 & Z = α 2 I - α β X + β 2 - α γ X 2 by putting the value of X   &   X 2 we get, Y = α β γ 0 α β 0 0 α   &   Z = α 2 - α β β 2 - α γ 0 α 2 - α β 0 0 α 2 We know that Y · Y - 1 = I ⇒ α β γ 0 α β 0 0 α 1 5 - 2 5 1 5 0 1 5 - 2 5 0 0 1 5