Consider the sequenee defined recursively by a1 2 an 2 ani

Consider the sequenee defined recursively by a_1 = -2, a_n = -2, an+i = -6a_n + 5a_n. We can use matrix diagonalization to lind an explicit formula for a_n. a. Find a matrix that satisfies b. Find the appropriate exponent k such that c. Find a diagonal matrix D and an invertible matrix P such that M = PDP^-1. g. You can find an explicit formula for a, using part b. and a formula for M^k = Try to develop this formula. Use your formula to verify the answer for part f.

Solution

Post one more question to get the remaining answer, i have solved the first four parts

a(n+1) = -6a(n-1) + 5an

a) The matrix M will have the first row as [0,1] and the second row will have the entries as [-6 5]

Hence the matrix will be

b) The value of k will be equal to (n-1), since the k term will be n-(n-1) = 1

Hence the value of k will be (n-1)

c) The diagonal matrix will be [2 0; 0 3] and the matrix P will be [1/2 1/3; 1 1]

d) Inverse matrix of P will be

[ 6 - 2; -6 3]

0	1
-6	5