If the matrix A = 1 0 0 0 2 0 3 0 - 1 satisfies the equation A 20 + α A 19 + β A = 1 0 0 0 4 0 0 0 1 for some real numbers α and β , then β - α is equal to ______.

Step-by-step solution

A = 1 0 0 0 2 0 3 0 - 1 A 2 = 1 0 0 0 4 0 0 0 1 , A 3 = 1 0 0 0 8 0 3 0 - 1 A 4 = 1 0 0 0 16 0 0 0 1 Hence, A 20 = 1 0 0 0 2 20 0 0 0 1 , A 19 = 1 0 0 0 2 19 0 3 0 - 1 So A 20 + α A 19 + β A = 1 + α + β 0 0 0 2 20 + α . 2 19 + 2 β 0 3 α + 3 β 0 1 - α - β = 1 0 0 0 4 0 0 0 1 Therefore α + β = 0 and 2 20 + 2 19 α - 2 α = 4 ⇒ α = 4 1 - 2 18 2 2 18 - 1 = - 2 Hence β = 2 So, β - α = 4