MAC 2103

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MAC 2103

• Module 3
• Determinants

2
Learning Objectives

• Upon completing this module, you should be able
  to
• Determine the minor, cofactor, and adjoint of a
  matrix.
• Evaluate the determinant of a matrix by cofactor
  expansion.
• Determine the inverse of a matrix using the
  adjoint.
• Solve a linear system using Cramers Rule.
• Use row reduction to evaluate a determinant.
• Use determinants to test for invertibility.
• Find the eigenvalues and eigenvectors of a matrix.

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Rev.F09
3
Determinants
There are three major topics in this module
Determinants by Cofactor Expansion Evaluating
Determinants by Row Reduction Properties of the
Determinant
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Rev.09
4
What is a Determinant?
A determinant is a real number associated with a
square matrix.
Determinants are commonly used to test if a
matrix is invertible and to find the area of
certain geometric figures.
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Rev.F09
5
How to Determine if a Matrix is Invertible?
The following is often used to determine if a
square matrix is invertible.
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Rev.F09
6
Example
Determine if A-1 exists by computing the
determinant of the matrix A. a) b) Solution a
) b)
A-1 does exist
A-1 does not exist
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Rev.F09
7
What are Minors and Cofactors?
We know we can find the determinants of 2 x 2
matrices but can we find the determinants of 3 x
3 matrices, 4 x 4 matrices, 5 x 5 matrices,
...? In order to find the determinants of larger
square matrices, we need to understand the
concept of minors and cofactors.
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Rev.F09
8
Example of Finding Minors and Cofactors
Find the minor M11 and cofactor A11 for matrix
A. Solution To obtain M11 begin by crossing out
the first row and column of A.
The minor is equal to det B -6(5) - (-3)(7)
-9
Since A11 (-1)11M11, A11 can be computed as
follows A11 (-1)2(-9) -9
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Rev.F09
9
How to Find the Determinant of Any Square Matrix?
Once we know how to obtain a cofactor, we can
find the determinant of any square matrix. You
may pick any row or column, but the calculation
is easier if some elements in the selected row or
column equal 0.
or
for any column j
for any row i
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Rev.F09
10
Example of Finding the Determinant by Cofactor
Expansion
Find det A, if Solution To find the
determinant of A, we can select any row or
column. If we begin expanding about the first
column of A, then det A a11A11 a21A21
a31A31. A11 -9 from the previous
example A21 -12 and A31 24
Now, try to find the determinant of A by
expanding the first row of A.
det A a11A11 a21A21 a31A31
(-8)(-9) (4)(-12) (2)(24) 72
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Rev.F09
11
How to Find the Adjoint of a Matrix?
The adjoint of a matrix can be found by taking
the transpose of the matrix of cofactors from A.
In our previous example, we have found the
cofactors A11, A21, A31. If we continue to solve
for the rest of the cofactors for matrix A,
namely A12, A22, A32 , A13, A23, and A33 , then
we will have a 3 x 3 matrix of cofactors from A
as follows
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Rev.F09
12
How to Find the Adjoint of a Matrix? (Cont.)
The transpose of this 3 x 3 matrix of cofactors
from A is called the adjoint of A, and it is
denoted by Adj(A). What are we going to do
with this Adj(A)? We can use it to help us find
the A-1 if A is an invertible matrix.
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Rev.F09
13
How to Find A-1 Using the Adjoint of a Matrix?
Theorem 2.1.2 If A is an invertible matrix,
then

• Note
• The square matrix A is invertible if and only if
  det(A) is not zero.
• If A is an n x n triangular matrix, then det(A)
  is the product of the entries on the main
  diagonal of the matrix (Theorem 2.1.3.)

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Rev.F09
14
What is Cramers Rule?
Cramers Rule is a method that utilizes
determinants to solve systems of linear
equations. This rule can be extended to a system
of n linear equations in n unknowns as long as
the determinant of the matrix is non-zero.
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Rev.F09
15
Example of Using Cramers Rule to Solve the
Linear System
Use Cramers rule to solve the linear
system. Solution In this system a1 1, b1
4, c1 3, a2 2, b2 9 and c2 5
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Rev.F09
16
Example of Using Cramers Rule to Solve the
Linear System (cont.)
E 7, F -1 and D 1 The solution is
Note that Gaussian elimination with backward
substitution is usually more efficient than
Cramers Rule.
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Rev.F09
17
What Are the Limitations on the Method of
Cofactors and Cramers Rule?

• The main limitations are as follow
• A substantial number of arithmetic operations are
  needed to compute determinants of large matrices.
• The cofactor method of calculating the
  determinant of an n x n matrix, n gt 2, generally
  involves more than n! multiplication operations.
• Time and cost required to solve linear systems
  that involve thousands of equations in real-life
  applications.
• Next, we are going to look at a more efficient
  method to find the determinant of a general
  square matrix.

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Rev.F09
18
Evaluating Determinants by Reducing the Matrix to
Row-Echelon Form
Just keep these in mind when A is a square
matrix 1. det(A)det(AT). 2. If A has a row of
zeros or a column of zeros, then det(A)0. 3. If
A has two proportional rows or two proportional
columns, then det(A)0.

• Let A be a square matrix. (See Theorem 2.2.3)
• If B is the matrix that results from scaling by
  a scalar k, then
• det(B) k det(A).
• (b) If B is the matrix that results from either
  rows interchange or columns interchange, then
  
• det(B) - det(A).
• (c) If B is the matrix that results from row
  replacement, then
• det(B) det(A).

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Rev.F09
19
How to Evaluate the Determinant by Row Reduction?
Lets look at a square matrix A. We can find
the determinant by reducing it into row-echelon
form. Step 1 We want a leading 1 in row 1. We
can interchange row 1 and row 2 to accomplish
this.
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Rev.F09
20
How to Evaluate the Determinant by Row Reduction?
(Cont.)
From Step 1
Step 2 We want a leading 1 in row 2. We can
take a common factor of 3 from row 2 to
accomplish this (scaling). Step 3 We want a
zero at both row 2 and row 3 below the leading 1
in row 1. We can add -3 times row 1 to row 3 to
accomplish this (row replacement).
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Rev.F09
21
How to Evaluate the Determinant by Row Reduction?
(Cont.)
Step 4 We want a zero below the leading 1 in row
2. We can add row 2 to row 3 to accomplish this
(row replacement). Step 5 We want a leading
1 in row 3. We take a common factor of -5/3 from
row 3 to accomplish this (scaling).
From Step 3
Remember If A is an n x n triangular matrix,
then det(A) is the product of the entries on the
main diagonal of the matrix.
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Rev.F09
22
Lets Look at Some UsefulBasic Properties of
Determinants

• Let A and B be n x n matrices and k is any
  scalar. Then,
• If A is invertible, then

Question Is det(AB) det(A) det(B) ?
Remember If A is an n x n triangular matrix,
then det(A) is the product of the entries on the
main diagonal of the matrix.
This is because A-1AI, det(A-1A) det(I) 1
det(A-1) det(A) 1, and so
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Rev.F09
23
What are Eigenvalues and EigenVectors?
An eigenvector of an n x n matrix A is a
nontrivial (nonzero) vector such that
, where is a scalar called an
eigenvalue. Linear systems of this form can be
rewritten as follows The system has a
nontrivial solution if and only if This is
the so called characteristic equation of A and
therefore B has no inverse, and the linear system
has infinitely many solutions.
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Rev.F09
24
Example
Express the following linear system in the form
Find the characteristic equation, eigenvalues
and eigenvectors corresponding to each of the
eigenvalues. The linear system can be written
in matrix form as
with
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Rev.F09
25
Example (Cont.)

which is of the form Thus,
Can
you tell what is the characteristic equation for
A?
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Rev.F09
26
Example (Cont.)
The characteristic equation for A is or

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Rev.F09
27
Example (Cont.)
Thus, the eigenvalues of A are By definition,
is an eigenvector of A if and only if is a
nontrivial solution of that is If ,
then we have Thus, we can form the
augmented matrix and solve by Gauss Jordan
Elimination.

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Rev.F09
28
Example (Cont.)
Lets form the augmented matrix and solve by
Gauss Jordan Elimination. Thus,
a free variable,

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Rev.F09
29
Example (Cont.)
Solving this system yields So the eigenvectors
corresponding to are the nontrivial solutions of
the form Similarly, if , then we
have

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Rev.F09
30
Example (Cont.)
Lets form the augmented matrix and solve by
Gauss Jordan Elimination. Thus,


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Rev.F09
31
Example (Cont.)
Solving this system yields So the eigenvectors
corresponding to are the nontrivial solutions of
the form

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Rev.F09
32
What have we learned?

• We have learned to
• Determine the minor, cofactor, and adjoint of a
  matrix.
• Evaluate the determinant of a matrix by cofactor
  expansion.
• Determine the inverse of a matrix using the
  adjoint.
• Solve a linear system using Cramers Rule.
• Use row reduction to evaluate a determinant.
• Use determinants to test for invertibility.
• Find the eigenvalues and eigenvectors of a matrix.

http//faculty.valenciacc.edu/ashaw/ Click link
to download other modules.
Rev.F09
33
Credit

• Some of these slides have been adapted/modified
  in part/whole from the text or slides of the
  following textbooks
• Anton, Howard Elementary Linear Algebra with
  Applications, 9th Edition
• Rockswold, Gary Precalculus with Modeling and
  Visualization, 3th Edition

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Rev.F09