# Maple Lab 6: Fibonacci and Lucas Sequenceswith(LinearAlgebra):# Problem 1.y0:=0:y1:=1:A:=<0,1|1,1>;u0:=<y0,y1>;<y5,y6> = (A^5).u0;# Problem 2.chareq:=CharacteristicPolynomial(A,lambda);
lambda1,lambda2:=solve(chareq,lambda);# Problem 3.# AP=PD verifies the eigenpairs. Check with Eigenvectors(A);P:=<-lambda2,1|-lambda1,1>;DD:=<lambda1,0|0,lambda2>;simplify(A.P-P.DD); # AP=PD verifies the eigenpairs# Problem 4.# Explain why A is diagonalizable.# Problem 5. Done in problem 3.# Problem 6.DD^10; <y10,y11> = simplify(P . DD^10 . (1/P) . u0);# AP=PD explains why A^10 = P D^10 (1/P)TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODg0MTQ5OTkwOTM4ODU0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMiIiEiIiJGJ0YnRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODg0MTQ5OTkwOTM4OTc0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjIiIhIiIiRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODg0MTQ5OTkwOTM5MDk0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjJSN5NUclI3k2R0YlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODg0MTQ5OTkwOTM5Njk0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjIiImIiIpRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODg0MTQ5OTkwODk3Nzc0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMsJiMhIiIiIiMiIiIqJCIiJiNGKkYpRi1GKiwmRitGJ0YnRipGKkYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODg0MTQ5OTkwODk4MTM0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMsJiokIiImIyIiIiIiI0YpRilGKiIiIUYsLCZGKUYqRicjISIiRitGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODg0MTQ5OTkwODk4NjE0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMiIiFGJkYmRiZGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODg0MTQ5OTkwODkyOTU4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMqJCwmKiQiIiYjIiIiIiIjRipGKkYrIiM1IiIhRi4qJCwmRipGK0YoIyEiIkYsRi1GJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODg0MTQ5OTkwODkzMDc4WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjJSR5MTBHJSR5MTFHRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODg0MTQ5OTkwODk0NjM4WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjIiNiIiMqKUYl