Homework 4
Due: Wednesday, September 24
  
   
1
1
2
4



 3   −1   0   1
   
 
1. Determine whether the set of vectors 
 6 , 1 , 6 , 2



4
2
0
10
dependent. If so, explain why. If not, show why not.
n
o
2. Let ~v ∈ Rn . Show that the set of vectors ~0, ~v is linearly dependent.

1
 2
3. Compute the the determinant of 
 1
0
1
0
1
1
0
1
2
0




 is linearly in



3
2 
.
1 
1
4. There is a connection between determinants and invertibility. The following two examples
will illustrate this fact.
4 1
(a) Compute both the determinant and the inverse (if it exists) of
.
1 3
3 1
(b) Compute both the determinant and the inverse (if it exists) of
.
6 2
5. Some problems in the real world involve rather small matrices, but many involve very large
matrices, with dimensions around 10, 1000, or even 1000000! One of the difficulties with
the computation of determinants is the computational complexity. In particular, for a 2×2
matrix, 3 flops (floating point operations) are needed to compute the determinant (two multiplications and one addition). For a 3×3 matrix, 14 flops are needed. This can be seen
by counting the number of flops needed at each step while computing the determinant by
moving along the first row.
Let Nk be the number of flops needed to compute the determinant of a k × k matrix.
(a) Prove that
Nk = ( Nk−1 + 2) · k − 1.
This could be part of an inductive proof...
(b) Using the recursive relation, how many flops are needed to compute the determinant
of a 10×10 matrix? This is actually a relatively small matrix...
Professor Dan Bates
Colorado State University
M369 Linear Algebra
Fall 2008