Answers to Gateway #11

Sec. 3.1 (BDH)

  (x') = (2 1) (x)
       (y')   (1 1) (y)

#17a Reducing to a first order system of equations by letting v = y', we get that v=0, and y = anything.

#17b Using the same approach we get the same answer.

#24 Just plug and chug.

Sec 3.2 (BDH)

#17a eigenvalues 2, -3, corresponding eigenvectors

   (0)  (0.98058067569092)
   (1)  (0.19611613513818)

#19a eigenvalues -5, -2, corresponding eigenvectors

   (-0.70710678118655)  (-0.44721359549996)
   (0.70710678118655 )  (-0.89442719099992)

#30 eigenvalues are a and d and corresponding eigenvectors

   (1)  (1      )
   (0)  ((d-a)/b)

#31 eigenvalues are 0.5*(a+d+ ( (a-d)^2+4b^2 )^(1/2) and 0.5*(a+d - ( (a-d)^2+4b^2 )^(1/2). Since the discrminant is positive these eigenvalues are always real. Doing some algebra one can see that the solutions are distinct so long as b is not zero.

Last modified: Thur Jan 30 16:44:48 1997