Name:
CSU ID:
Homework 2
February 2, 2015
1. S1.5 ]8
2. S1.5 ]10
3. Determine the value λ for which the following matrix is not invertible.


1 −2 −1


4
λ 
 1
4 −5
1
4. Determine the values of λ for which the following matrix is not invertible.


λ 0 1


 −1 λ 2 
2 0 8
5. (a) Show that the matrix is orthogonal(i.e., AT A = AAT = I).


cos θ 0 − sin θ


1
0
A= 0

sin θ 0 cos θ
T
(b) For ~x = [1, −2,
x|| and ||A~x||. Note: If ~y = [y1 , y2 , y3 ]T
q 3] , find ||~
then ||~y || =
y12 + y22 + y32 .
6. Consider the linear system



2x + y =
3
−x − y = −2

 3x + 2y =
5
(a) Write the linear system in matrix form A~x = ~b and find the
solution.
(b) For most systems of the same size as the one given, does a solution
exist?
(b) Find a 2 × 3 matrix C such that CA = I. Hint: Since C is 2 × 3,
there are 6 elements to solve for. Pose CA = I as a system of
equations such that the unknowns are the elements of C.
(c) Show that the solution to the linear system is given by ~x = C~b.
7. Consider the matrix A given below. Find A−1 by finding the seven
elimination matrices needed. Note: The matrix E2 would eliminate
everything below the (2,2) position, not just one term. There are three
elimination matrices needed to bring A to upper triangular form, one
to force the diagonal positions to become 1, and three to eliminate the
positions above the diagonal.




A=
2
6 −8 −2
−4 −13
11
3
8
27 −13 −21
14
40 −70 −5





8. (a) Find the LU decomposition for the matrix A given in the previous problem. The information from the previous problem should
make this doable with no additional work.
(b) Solve A~x = ~b = [−2, −3, 1, −21]T by doing the forward solution
of L~y = ~b followed by the backward solution of U~x = ~y . These
solutions can be done on a calculator.