Name:
CSU ID:
Homework 3
September 11, 2013
Show work to receive credit.
1. Determine the values of a, b, c such that A is symmetric.


−2 4 5


A= a 1 5 
b c 7
2. Determine the value λ for which the following matrix is not invertible.


1 −2 −1


4
λ 
 1
4 −5
1
3. Determine the values of λ for which the following matrix is not invertible.


λ 0 1


 −1 λ 2 
2 0 8
"
4. Find the inverse of A =
cos θ sin θ
− sin θ cos θ
#
5. Consider the matrix A below. Compute (AT )−1 and (A−1 )T . How are
they related?


1 −1 −2


 2 −3 −5 
−1
3
5
6. (NOT on quiz. Could be on exam.) 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. Given A, find
(a) Determine E1 , E2 , and E3 where Ej is the elimination matrix
that zero out the elements in column j of A below the diagonal
j, j position A as discussed in class.
(b) Determine the upper triangular matrix U = E3 E2 E1 A.
(c) Find L = E1−1 E2−1 E3−1 . Do the inverses actually need to be
computed are is the information already known?
(d) Using forward and backward substitution (in that order), solve
A~x = ~b = [−3, −2, −59, 116]T by hand. Your answer can be
checked on a calculator.




A=
2
1 −3
1
12 11 −17 12
14 −3 −27 −3
2 26
18 28



,



~b = 


−3
−2
−59
116





8. Suppose that A is a square 3 matrix with an LU factorization such
that diagonal elements of L are 1. Show that A is invertible if and
only if the diagonal elements of U are nonzero.