Let 2x2 matrix M=[-5 2/ -2 -5]. Find formulas for the entries of M^n, where n is a positive integer.Start by diagonalizing M, i.e. find Q and D such that Q^(-1)MQ=D. Then M=QDQ^(-1), and so M=QD^nQ^(-1). Note that you will get complex eigenvalues, but since the entries of M are real, the entries of M^n must also be real. You might find it helpful to write complex numbers in polar form re^(ia), and to use the formula e^(ia)=cos(a)+isin(a).

The 2×22×2 matrixA=(5−53−3)𝐴=(53−5−3)  has two distinct real eigenvalues.  1. Give the characteristic polynomial  for  A𝐴  in Maple notation in the form   t^2 + a*t + b          Characteristic polynomial  =       2. Find the set of eigenvalues for  A𝐴 . Enter your response in Maple set notation, i.e., enclosed in braces { } with the two eigenvalues separated by a comma. A typical answer might look like {-4, 7}.     The set of eigenvalues for  A𝐴 is       3. Find one eigenvector for each eigenvalue, using Maple notation for vectors, e.g.  <5, -6> for the vector  (5−6)(5−6)        An eigenvector for the smallest eigenvalue for  A𝐴 is            An eigenvector for the largest eigenvalue for  A𝐴 is

Matrix A has characteristic polynomial p(𝜆) = (−45 − 𝜆)(5 − 𝜆)(27 − 𝜆).A = −3 4 32 4 3 −1632 −16 −13Given this information, find the spectral form of A, i.e. A = QΛQT

Use the quadratic formula to solve x2 – 5x – 2 = 0.A.B.C.D.

Determine whether or not the 5x5 matrix 𝐴=(3 0 -5 -6 2/ 6 2 -12 -9 3/ 6 0 -10 -9 3/ -5 0 7 5 -1/ 1 0 -5 -6 4) is diagonalizable. You may use the fact that the characteristic polynomial is 𝑐𝐴(𝑡)=(𝑡−2)^3(𝑡+1)^2.

Find a 2×22×2 matrix such that[−34−1−5][−3−14−5] ⎡⎣⎢⎢⎢⎢[ ⎤⎦⎥⎥⎥⎥] =[1001