Let α and β be real numbers. Consider a 3 × 3 matrix A such that A 2 = 3 A + α I . If A 4 = 21 A + β I , then

Step-by-step solution

Given, A 2 = 3 A + α I ∴ A 3 = A 2 . A ⇒ A 3 = 3 A 2 + α A ⇒ A 3 = 3 3 A + α I + α A ⇒ A 3 = 9 A + α A + 3 α I Now, A 4 = A 3 . A A 4 = 9 + α A 2 + 3 α A = 9 + α 3 A + α I + 3 α A = A 27 + 6 α + α 9 + α . . . . . . . 1 Given, A 4 = 21 A + β I . . . . . . . . . 2 On comparing equation 1 and equation 2 , ⇒   27 + 6 α = 21 ⇒ α = - 1 ⇒ β = α 9 + α = - 8