Math 317: Linear Algebra
Homework 5
Due: October 8, 2015
The following problems are for additional practice and are not to be turned in: (All
problems come from Linear Algebra: A Geometric Approach, 2nd Edition by ShifrinAdams.)
Exercises: Section 2.4: 1,4,6,8,14(a)
Section 2.5: 1,2,4–8,11,13,15
Turn in the following problems.
1. Section 2.5, Problem 12
2. Section 2.5, Problem 16
3. Section 2.5, Problem 22
4. Section 2.5, Problem 23
5. Section 2.5, Problem 24
6. Suppose that I is an n × n identity matrix, and u, v ∈ Rn . Recall that elementary
matrices are matrices of the form I − uvT . Prove that any elementary matrix is
invertible and the inverse is once again an elementary matrix. Hint: Show directly
−1
uvT
.
that I − uvT
=I− T
v u−1


1 4 5
7. Let A = 4 18 26.
3 16 30
(a) Determine the LU factors of A.
(b) Use the LU factors to solve Ax1 = b1 as well as Ax2 = b2 where


6
b1 =  0 
−6
 
6
and b2 =  6  .
12
8. Prove or give a counterexample (You must show that your counterexample works!):
Every n × n nonsingular matrix has an LU factorization. Hint: See Problem 14,
part a in Section 2.4.
1