week09 - Math 43, Fall 2007 B. Dodson Week 9: Monday:...

Math 43, Fall 2007
B. Dodson
Week 9:
Monday: Finish Suggested Hw8, start on Hw9 material
1. Determinants
2. Properties of Dets
3. Eigenvalues and Eigenvectors (2-by-2 in 4.1;
then n-by-n in 4.3)
——–
We compute det


2 1 5
4 2 3
9 5 1


using the
(first) row expansion (by minors):
det ( A ) = 2
fl
fl
fl
fl
2 3
5 1
fl
fl
fl
fl
- 1
fl
fl
fl
fl
4 3
9 1
fl
fl
fl
fl
+ 5
fl
fl
fl
fl
4 2
9 5
fl
fl
fl
fl
= 2(2 - 15) - (4 - 27) + 5(20 - 18)
= 2( - 13) - ( - 23) + 5(2) = - 26 + 33 = 7 .
———–
Problem
Reduce A =



2 1 3 5
3 0 1 2
4 1 4 3
5 2 5 3



to an upper triagular
matrix and use the reduction to find det ( A ) .
Solution:
A →



- 1 1 2 3
3 0 1 2
4 1 4 3
5 2 5 3



( r
1
→ r
1
- r
2
)
→



- 1 1 2 3
0 3 7 11
0 5 12 15
0 7 15 18



( r
2
→ r
2
+ 3 r
1
, r
3
→ r
3
+ 4 r
1
, r
4
→ r
4
+ 5 r
1
)

2
→



- 1 1 2 3
0 3 7 11
0 5 12 15
0 1 1 - 4



( r
4
→ r
4
- 2 r
2
)
——–
→



- 1 1 2 3
0 1 1 - 4
0 5 12 15
0 3 7 11



( r
2
↔ r
4
)
→



- 1 1 2 3
0 1 1 - 4
0 0 7 35
0 0 4 23



( r
3
→ r
3
- 5 r
2
, r
4
→ r
4
- 3 r
2
)
→



- 1 1 2 3
0 1 1 - 4
0 0 - 1 - 11
0 0 4 23



( r
3
→ r
3
- 2 r
4
)
→



- 1 1 2 3
0 1 1 - 4
0 0 - 1 - 11
0 0 0 - 21



= U ( r
4
→ r
4
+ 4 r
3
).
Now we have det U = ( - 1)(1)( - 1)( - 21) = - 21 ,
and det A = ( - 1) det U = 21 , since all of the 3rd
elementary operations change the determinant by a
factor of (1) – so no change - there are no 2nd
EROs and exactly one 1st ERO, with each 1st ERO
changing the determinant by a factor of ( - 1) .
Notice that, for the problem of finding the value of
the determinant, we do not need a reduced row echelon
or 1’s on the diagonal, and stop at a triangular form.

3
3. Eigenvalues and Eigenvectors
—————
A vector v = 0 in R
n
is an eigenvector with eigenvalue λ of
an n -by- n matrix A if Av = λv.
We re-write the vector equation as ( A - λI
n
) v = 0 ,
which is a homogeneous system with coef
matrix ( A - λI ) , and we want λ
so that the system has a non-trivial solution.