The Determinant Function
MATH 322, Linear Algebra I
J. Robert Buchanan
Department of Mathematics
Spring 2007
J. Robert Buchanan The Determinant Function
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Determinant Function in Linear Algebra I, Study notes of Linear Algebra
An excerpt from a university textbook on linear algebra i, specifically from the chapter on the determinant function. It explains the concept of a determinant as a function that takes a square matrix as an input and returns a scalar real number as an output. The determinant of a 2x2 matrix as an example and then extends the definition to nxn matrices using the concept of permutations and signed elementary products. The document also includes instructions for students to work on exercises related to this topic.
Typology: Study notes
Pre 2010
1 / 24
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Determinant Function in Linear Algebra I and more Study notes Linear Algebra in PDF only on Docsity!
The Determinant Function
MATH 322, Linear Algebra I
J. Robert Buchanan
Department of Mathematics
Spring 2007
Introduction
Basic Idea: a determinant is a function, with specific properties we will mention later, whose domain is a set of square matrices and whose range is a set of scalar, real numbers
Example
If A =
[
a b c d
]
, then det( A ) = ad − bc.
General Matrices
Now we develop the notion of a determinant for n × n matrices.
Definition A permutation of the set of integers { 1 , 2 ,... , n } is any rearrangement without omissions or repetitions.
Example If we consider { 1 , 2 , 3 } there are 6 permutations:
( 1 , 2 , 3 ), ( 1 , 3 , 2 ), ( 2 , 1 , 3 ), ( 2 , 3 , 1 ), ( 3 , 1 , 2 ), ( 3 , 2 , 1 )
Remark: There are n! permutations of n objects.
General Matrices
Now we develop the notion of a determinant for n × n matrices.
Definition A permutation of the set of integers { 1 , 2 ,... , n } is any rearrangement without omissions or repetitions.
Example If we consider { 1 , 2 , 3 } there are 6 permutations:
( 1 , 2 , 3 ), ( 1 , 3 , 2 ), ( 2 , 1 , 3 ), ( 2 , 3 , 1 ), ( 3 , 1 , 2 ), ( 3 , 2 , 1 )
Remark: There are n! permutations of n objects.
Inversions
Notation: a permutation of { 1 , 2 ,... , n } will be written as ( j 1 , j 2 ,... , jn ). Any permutation can be expressed as the product of transpositions of pairs of integers.
Definition An inversion has occurred if a larger integer precedes a smaller integer.
Inversions
Notation: a permutation of { 1 , 2 ,... , n } will be written as ( j 1 , j 2 ,... , jn ). Any permutation can be expressed as the product of transpositions of pairs of integers.
Definition An inversion has occurred if a larger integer precedes a smaller integer.
Counting Inversions
The total number of inversions in a permutation ( j 1 , j 2 ,... , jn ) can be determined as follows: (^1) Count the number of integers following j 1 that are smaller than j 1. (^2) Add to that the number of integers following j 2 that are smaller than j 2. (^3) Continue for j 3 , j 4 ,... , jn − 1.
Example Count the number of inversions in ( 4 , 3 , 2 , 5 , 6 , 1 ).
Parity
Definition A permutation is called even if the total number of inversions is even, and is called odd if the total number of inversions is odd.
Example Determine the parities of the following two permutations. (^1) ( 1 , 3 , 4 , 2 , 5 ) (^2) ( 3 , 4 , 5 , 2 , 1 )
Elementary Product
Definition An elementary product from an n × n matrix A is any product of n entries from A such that no two factors come from the same row or column.
Example List all the elementary products of the following matrices. 1
[
a 11 a 12 a 21 a 22
]
2
a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33
Observation: there are n! elementary products of an n × n matrix.
Elementary Product
Definition An elementary product from an n × n matrix A is any product of n entries from A such that no two factors come from the same row or column.
Example List all the elementary products of the following matrices. 1
[
a 11 a 12 a 21 a 22
]
2
a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33
Observation: there are n! elementary products of an n × n matrix.
Signed Elementary Products
Think of an elementary product written as a 1 j 1 a 2 j 2 · · · anjn where ( j 1 , j 2 ,... , jn ) is a permutation of the set { 1 , 2 ,... , n }. Definition A signed elementary product from A is an elementary product a 1 j 1 a 2 j 2 · · · anjn multiplied by +1 if ( j 1 , j 2 ,... , jn ) is an even permutation, or −1 if ( j 1 , j 2 ,... , jn ) is an odd permutation.
Example List all the signed elementary products of the following matrices. [ a 11 a 12 a 21 a 22
]
a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33
Signed Elementary Products
Think of an elementary product written as a 1 j 1 a 2 j 2 · · · anjn where ( j 1 , j 2 ,... , jn ) is a permutation of the set { 1 , 2 ,... , n }. Definition A signed elementary product from A is an elementary product a 1 j 1 a 2 j 2 · · · anjn multiplied by +1 if ( j 1 , j 2 ,... , jn ) is an even permutation, or −1 if ( j 1 , j 2 ,... , jn ) is an odd permutation.
Example List all the signed elementary products of the following matrices. [ a 11 a 12 a 21 a 22
]
a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33
Determinant Function
Definition Let A be an n × n matrix, the determinant of A , denoted det( A ) is the sum of all the signed elementary products of A.
Example Find the determinants of the following matrices. 1
[
a 11 a 12 a 21 a 22
]
2
a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33
Evaluating 3 × 3 Determinants
det
a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33
a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33
a 11 a 12 a 21 a 22 a 31 a 32
Add all the elementary products pointing to the right and subtract all the elementary products pointing to the left. Warning: mnemonic device works only for 2 × 2 and 3 × 3 matrices.