Can someone help with these questions

Question 1)

3 Given A = has eigenvectors 12 -J V and v2 = 2 3 The respective eigenvalues are then A1 = 1 and 12 = Number So we can write 0 A =M M-1 0 where M = . Using this we can calculate that the bottom left entry of A2015 is a 2015 + 6 2015 2 where 1 a = Number and Number 1 2 Recall: the Maple notation for the matrix is , >. 3 4We can use the factorisation -5 24 -1 A = -2 9 1 3 a b to obtain explicit formulas for the entries of All = in terms of n . Doing so gives C d b = C= and d =What relationship, if any, is there between the eigenvectors of a matrix A and the eigenvectors of its transpose A/ ? Let's investigate. Consider the matrix A = 7 2 -6 14 , with eigenvalues 1 = 10 and 12 = Number and with associated eigenvectors = and v2 = Using this information we can write the associated diagonal form A - (3 2 ) (10 1 ) ( 3 21 ) Now recall that for 2 x 2 matrices R and S , O (RS)T _ RT ST O (RS) T _ RTS-1 O (RS)T = S-1RT O (RS) T _ STRT. Thus, taking the transpose of the matrix product above gives the following matrix product 10 0 11 Recall: the Maple notation for the matrix 2 3 is , >. It follows that A also has eigenvalues ] = 10 and 12 = 11 , with associated eigenvectors W1 = and w2 = Recall: the notation for the vector 2 is . Can you deduce the general rule?A (non-traditional) Fibonacci sequence is ag = 1 , a1 = 1 , a2 = 2 , ag = 3 , a4 =5 etc, an with an +2 = anti + an for n 2 0 . Let Vn = for n = 0, 1, 2, 3, . . .. Then an+1 the Fibonacci pattern is contained in the matrix equation Vn+1 = Avn where A is the 2 x 2 matrix A = Note: the Maple notation for the matrix 1 2 3 is , >. Since v1 = Avo , V2 = Av] = Advo etc, we see that Vn = Anvo = An Let's use this to find a formula for an. The characteristic polynomial of A is p(t) = det(A - tI) = This equation has two famous zeroes; the smaller of which is 1 = Number and the larger of which is 12 = Number The corresponding eigenvectors are w1 = ( m ) and w2 - where 17 = Number and 72 = Number So if we set M = 1 then A = MDM - where D = 0 0 12 Working through this, we get the formula for an . Namely an = Note: that the Fibonacci sequence traditionally starts from ag = 0, a] = 1 , so your answer may not be exactly the same as one found in a text book