Load the worksheet in 
  f:\math115\11507mid.mws 
and use the supplied matrices to check the solution of the first 
3 questions of the midterm examination which is stored at:
 f:\math115\11507mid.pdf
(you can use any commands that we have used so far)

Given this matrix
> F:=Matrix([[1, -8, -15], [8, -19, -21], [-6, 12, 12]]);
 find its eigenvalues and an eigenvector 
using row and column operations and check your answer with 
> Eigenvectors(F);
and identify your eigenvector in the answer.

Form the diagonalisation matrices using the command DiagonalMatrix
and check that FP=PD, recalling that we cannot use D, so use DM, say.

Create DP as the kth power of DM using the individual elements
and check that DM^3 is the same as subs(k=3,DP);
Verify that F^2 does not contain the squares of the elements of F 
because F is not a diagonal matrix.

Form a matrix FP using the diagonalisation power formula.
Check that when you substitute k as -1, 0, 1 and 3 you get the expected matrix.

Given that a(n) = 5*a(n-1) +14*a(n-2), a(0)=-1 and a(1)=3, form a matrix
which represents this recurrence and find a general solution for a(j)

Check your answer with
> rsolve({a(n) = 5*a(n-1) +14*a(n-2),a(0)=-1,a(1)=3}, a(j));