Today

1
Lecture 5

• Today
• Evaluation of a Determinant using E. O.
• Properties of Determinants
• Introduction to Eigenvalues
• Applications
• Reading Assignment Secs 3.3 - 3.5 of Textbook
• Homework 3 assigned

Elementary Linear Algebra Larsen Falvo (6th
Edition) TKUEE???-NTUEE SCC_10_2008
2
Lecture 5

• Next Time
• Vectors in Rn
• Vector Spaces
• Subspaces of Vector Spaces
• Reading Assignment Secs 4.1 - 4.3 of Textbook

Elementary Linear Algebra Larsen Falvo (6th
Edition) TKUEE???-NTUEE SCC_10_2008
3
What have you learned about E.O. and Determinants?

• Q1 Why can every invertible matrix be
  represented by mutiplications of elementary
  matrices?
• Q2 How is determinant related to systems of
  linear equations?

4
Geometry of Determinants Determinants as Size
Functions

• We have so far only considered whether or not a
  determinant is zero, here we shall give a meaning
  to the value of that determinant.

One way to compute the area that it encloses is
to draw this rectangle and subtract the area of
each subregion.
5

• The properties in the definition of determinants
  make reasonable postulates for a function that
  measures the size of the region enclosed by the
  vectors in the matrix.
• See this case

6

• Another property of determinants is that they are
  unaffected by pivoting. Here are before-pivoting
  and after-pivoting boxes (the scalar used is
• k 0.35).

Although the region on the right, the box formed
by and , is more slanted than
the shaded region, the two have the same base and
the same height and hence the same area.
This illustrates that
7

• That is, weve got an intuitive justification to
  interpret det ( , . . . , ) as the
  size of the box formed by the vectors.

Example The volume of this parallelepiped, which
can be found by the usual formula from high
school geometry, is 12.
8

• The only difference between them is in the order
  in which the vectors are taken. If we take
  first and then go to , follow the
  counterclockwise are shown, then the sign is
  positive. Following a clockwise are gives a
  negative sign. The sign returned by the size
  function reflects the orientation or sense of
  the box.

9

• Volume, because it is an absolute value, does not
  depend on the order in which the vectors are
  given. The volume of the parallelepiped in the
  following example, can also be computed as the
  absolute value of this determinant.

The definition of volume gives a geometric
interpretation to something in the space, boxes
made from vectors.
10
3.2 Evaluation of a determinant using elementary
operations

• Thm 3.3 (Elementary row operations and
  determinants)

Let A and B be square matrices.
11

• Ex

12
Note A row-echelon form of a square matrix
is always upper triangular.

• Ex 2 (Evaluation a determinant using
  elementary row operations)

13

• Notes

14

• Notes

15

• Thm 3.4 (Conditions that yield a zero
  determinant)

If A is a square matrix and any of the following
conditions is true, then det (A) 0.
(a) An entire row (or an entire column) consists
of zeros.
(b) Two rows (or two columns) are equal.
(c) One row (or column) is a multiple of another
row (or column).
16

• Ex

17

• Note

Number of operations for cofactor expansion of
nxn matrix n! 30! ?
18

• Ex 5 (Evaluating a determinant)

Sol
19

• Ex 6 (Evaluating a determinant)

Sol
20
(No Transcript)
21
3.3 Properties of Determinants

• Thm 3.5 (Determinant of a matrix product)

det (AB) det (A) det (B)

• Notes

(1) det (EA) det (E) det (A)
(2)
(3)
22

• Ex 1 (The determinant of a matrix product)

Find A, B, and AB
Sol
23

• Check

AB A B
24

• Thm 3.6 (Determinant of a scalar multiple of a
  matrix)

If A is an n n matrix and c is a scalar, then
det (cA) cn det (A)

• Ex 2

Find A.
Sol
25

• Thm 3.7 (Determinant of an invertible matrix)

A square matrix A is invertible (nonsingular) if
and only if det (A) ? 0

• Ex 3 (Classifying square matrices as singular
  or nonsingular)

Sol
A has no inverse (it is singular).
B has inverse (it is nonsingular).
26

• Thm 3.8 (Determinant of an inverse matrix)

• Thm 3.9 (Determinant of a transpose)

• Ex 4

(a)
(b)
Sol
27

• Equivalent conditions for a nonsingular matrix

• If A is an n n matrix, then the following
  statements are equivalent.

(1) A is invertible.
(2) Ax b has a unique solution for every n 1
matrix b.
(3) Ax 0 has only the trivial solution.
(4) A is row-equivalent to In
(5) A can be written as the product of
elementary matrices.
(6) det (A) ? 0
28

• Ex 5 Which of the following system has a unique
  solution?

(a)
(b)

29

• Sol

(a)
This system does not have a unique solution.
(b)
This system has a unique solution.
30
3.4 Introduction to Eigenvalues

• Eigenvalue problem

If A is an n?n matrix, do there exist n?1 nonzero
matrices x such that Ax is a scalar multiple of
x?
31

• Ex 1 (Verifying eigenvalues and eigenvectors)

32

• Question
• Given an n?n matrix A, how can you find
  the eigenvalues and
• corresponding eigenvectors?

33

• Ex 2 (Finding eigenvalues and eigenvectors)

34
(No Transcript)
35

• Ex 3 (Finding eigenvalues and eigenvectors)

Sol Characteristic equation
36
(No Transcript)
37
(No Transcript)
38
3.5 Applications of Determinants

• Matrix of cofactors of A

• Adjoint matrix of A

39

• Thm 3.10 (The inverse of a matrix given by its
  adjoint)

If A is an n n invertible matrix, then

• Ex

40

• Ex 2

(a) Find the adjoint of A.
(b) Use the adjoint of A to find
Sol
41
cofactor matrix of A
42

• Thm 3.11 (Cramers Rule)

(this system has a unique solution)
43
( i.e.
)
44

• Pf

A x b,
45
(No Transcript)
46

• Ex 6 Use Cramers rule to solve the system of
  linear equations.

Sol
47
Keywords in Section 3.5

• matrix of cofactors ?????
• adjoint matrix ????
• Cramers rule Cramer ??

48

• Today
• Inverse of a Matrix
• Elementary Matrices
• Determinant of a Matrix
• Evaluation of a Determinant using Elementary
  Operations
• Reading Assignment Secs 2.5 3.1-3.2 of
  Textbook
• Homework 2 Due and 3 Assigned
• Next Time (10/24, Class 330pm 620pm)
• Evaluation of a Determinant using E. O. (Cont.)
• Properties of Determinants
• Introduction to Eigenvalues
• Applications
• Reading Assignment Secs 3.2 - 3.5 of Textbook