Let A = [-8 15 15 -8 13 10 3 -5 -4]. (a) Use MATLAB to compute A^n for n = 2, 3, 4, 5, 6, 7, 8. Do you notice a pattern? (b) Have MATLAB produce an invertible P and a diagonal D such that A = PDP^-1, Notice that complex numbers get involved. (c) To understand A^n, it suffices to understand D^n because A^n = PD^n P^-1. Describe the pattern that emerges when we consider powers of D: D, D^2, D^3, D^4, etc. (d) Without doing a computation in MATLAB, determine A^1000001

## Expert Answer

a)

A^{n*4}=I (Identity matrix) where n=1,2,3. and so on

so A^{n*4+1} =A where n=1,2,3….and so on

b)

c)

we can observe that D^{4} = I (Identity matrix)

and the pattern is D^{4*n} = I where n=1,2,3 … and so on

D^{4*n+1} = D where n=1,2,3 … and so on

d) To determine A^{10000001} by 4 divisibility rule we know that 10000001 gives remainder 1

so it is in the pattern A^{4*n+1} which is equal to A

so A^{10000001} = A