Chapter 4
Determinant
4.1 Definition and Properties of the Determinants
4.2 Cofactor Expansion and Applications of the Determinants
4.3 Cramer's Rule
*4.4 Application of Determinant
4.5 Eigenvalues and Eigenvectors
(English Textbook) http://matrix.skku.ac.kr/2015-Album/Big-Book-LinearAlgebra-Eng-2015.pdf
(e-book : Korean) http://matrix.skku.ac.kr/2015-Album/BigBook-LinearAlgebra-2015.pdf
http://matrix.skku.ac.kr/LA-Sage/
선형대수학 http://matrix.skku.ac.kr/LinearAlgebra.htm
Credu http://matrix.skku.ac.kr/Credu-CLA/index.htm
OCW http://matrix.skku.ac.kr/OCW-MT/index.htm
행렬 계산기 http://matrix.skku.ac.kr/2014-Album/MC-2.html
그래프 이론 http://matrix.skku.ac.kr/2014-Album/Graph-Project.html
행렬론 http://matrix.skku.ac.kr/MT2010/MT2010.htm
JFC http://matrix.skku.ac.kr/JCF/index.htm
The concept of determinant was introduced 150 years before the use of modern matrix, and we have used the determinant to solve the systems of linear equations for over 100 years. In late 19th century, Sylvester introduced the concept of matrix and the method for solving systems of equations by using an inverse matrix, where the determinant is used to check if an inverse of a matrix exists or not. Also, the determinant can be used to find area, volume, equations of lines or planes, and exterior product. It also helps in geometric interpretation of vectors.
In this chapter, we first define the determinant and review its properties. Then we study how to compute the determinant by cofactor expansion. We also study Cramer's rule which solves the systems of linear equations by using the determinant.
One of the most important concepts in linear algebra is eigenvalues and eigenvectors. Eigenvalues have almost all important informations by 
components from an object with 
components. Eigenvalues are not only important in theoretical perspective but also applicable to almost all areas related to matrix, such as, finding the solutions of differential equations, computing the power of given matrix, Google search, and image compression, etc. In the last section of this chapter, we compute eigenvalues by using the determinant.
4.1 Definition of Determinant
Reference video: https://youtu.be/MeVCFwg1Eq0 (http://youtu.be/DM-q2ZuQtI0)
Practice site: http://matrix.skku.ac.kr/knou-knowls/CLA-Week-5-Sec-4-1.html
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In this section, we introduce a determinant function which assigns any square matrix |
to a real number 
. In order to define the determinant function, we first introduce
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Definition |
[Permutation] |
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For a set of natural numbers
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, 
to 
We simply denote a permutation as 
. As a permutation 
is an one to one correspondence, the range 
is simply a rearrangement of 
. Hence, there are 
permutations on 
: fixing the position of first element leaves 
possibilities to permute the rest. We denote the set of all permutations of set 
by 
.
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[Remark] |
Inversion |
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In permutation
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, an 
, 
is placed on the left hand side of 
, and hence 
is an inversion. Similarly, 
is an inversion.
Number of inversions for 
: after (
)-th index, the number of indexes which is smaller than 
-th index 
is called the number of inversions for 
. In the above example, the number of inversion for 
is 
. Number of inversions for a permutation 
is the total sum of each number of inversions for 
, 
.
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Definition |
[Even permutation and odd permutation] |
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If number of inversions for a permutation is even than it is called an even permutation, If the number is odd than it is called an odd permutation. |
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Determine whether the permutation is even or odd by computing the inversion numbers for a permutation 
in 
.
The number of inversions for 5 is 4. The number of inversions for 1 is 0, for 2 is 0, for 4 is 1, and 3 is the last index. Hence, the total sum is 
, and it is an odd permutation. □
● http://matrix.skku.ac.kr/RPG_English/4-TF-Permutation.html
, which assigns each permuta tion of 
to either +1 or -1 as follows.
to either even or odd permutation.
In permutation, if two numbers switch the location then the signature is changed
be a permutation by switching any two numbers from given permutation 
. Then
be an 
matrix. We denote the determinant of matrix 
as 
or 
and define it as follows. 
By definition, 
matrix 
has it's determinant as 
.
Each term 
in the determinant is from the matrix 
, by choosing a row and a column, without any overlapping, then multiplying them and assigning a corresponding signature.
, where 
.
is 
matrix, 
. Since 
, 
, we have
. 
, where 
.
is 
matrix,
.
, 
, 
, 
,
, 
,
, 
.
.
. □
terms, and for degree 10, there are 
terms to compute)
and its transpose matrix 
have the same determinant. 
, 
, and 
. Since
. □
be a matrix obtained by switching two rows (columns) from a square matrix
then 
.
Let 
be a matrix obtained by replacing 
th and 
th row of 
, 
. This means 
, 
and 
if 
.
(by definition)
(by theorem 4.1.1)
. Since 
and 
, 
. 
has two identical rows (columns) then 
.
which has identical first and third rows. Note
has a row (column) with identical zeros then 
.
which has identical zeros in the third row. Observe
be a matrix obtained by multiplying 
times a row of a square matrix 
. Then 
. Note that
has two proportional rows then 
.
be a square matrix and 
times of one row is added to another row of 
and name this new matrix as 
, then 
.
Let 
be a new matrix whose second row is obtained by adding 
times of the first row of 
to 
.
. (by Theorem 4.1.4) 
and 2 times of the second row is added to the first row and name it as matrix 
. Then 
. 
. 
is an 
triangular matrix, the determinant of 
equals the product of the diagonal elements. That is, 
. 
, where
be an 
.
is obtained by multiplying 
to a row of 
, 
is obtained by switching two rows of 
, 
is obtained by multiplying 
times a row and adding it to another row of 
, 
is an 
matrix and 
is an elementary matrix, 
.
is invertible if and only if 
.
matrices 
and 
, 
.
.
Since 
, and
. 
is invertible then 
and 
.
.
is invertible with 
. Observe 
and
. 
Reference Video: 
Practice Site: 
th row and 
th column of a given square matrix 
, as 
. We call its determinant 
as 
for 
. We also call 
as 
for 
.
, find the minor and cofactor of 
for 
.
for 
is 
and the cofactor of A for 
is 
.
be a cofactor of 
matrix 
for 
. The matrix 
is called an 
and is denoted by adj
. That is,
of the following matrix.
for each element is as follows.
matrix 
can be expanded as follows.
For any 
matrix 
, the following identity holds. That is,
.
.
,
be an 
matrix. For any 
(
) the following holds.
(cofactor expansion 
th row)
(cofactor expansion 
th column)
When computing the determinant, it is advantageous to use the cofactor 
.
. 
be an 
invertible matrix, then the inverse matrix of 
is
.
.
Reference video:
Practice site: 
Cramer's rule can be applied to 
be a coefficient matrix, and 
. Then the system of linear equations can be written as 
. If 
, the system of linear equations has a unique solution as follows:
denotes the matrix 
with 
th column replaced by the vector 
.
Since
, 
is invertible. Hence the system of linear equations 
has a unique solution 
. Since 
, we have
.
th component of 
is 
. Since
,
as a matrix 
with 
th column replaced by the vector 
, then we have
. 
Let 
be the coefficient matrix. Then
, and hence
,
, 
.
From 
, and each matrix
.
matrix 
, the following are equivalent.
is a product of elementary matrices.
is invertible.
is the unique solution to 
.
has a unique solution for any 
.
are linearly independent.
are linearly independent.
Note that there are more equivalent statements for the above theorem. For more equivalent statements, refer Theorem 7.4.9 in section 7.4. 
Reference video: 
Practice site: 
By using a determinant, we can easily find 
, is as follows.
and 
. As the equation holds by substituting 
and 
into the equation, the equation must be the equation of the line which passes through two given points.
■
Similar to 
, an equation of a plane, which passes through three distinct points 
, is as follows:
.
Consider an arbitrary non-singular square matrix 
. Let 
be the 
th column and
.
is a 
is a generalized 
Parallelogram can be expressed by adding two vectors as follows.
which is the same as the absolute value of 
. Similarly, a parallelepiped is generated by three vectors which do not lie on the same plane. Let matrix 
's columns consist by these three vectors. Then the volume of the parallelepiped is absolute value of 
.
be an 
matrix. The 
.
be an 
matrix. The 
.
, is 
, where 
.
We will prove only (3).
.
. Now, the determinant
.
, 
, 
is as follows.
and 
. Hence, by Theorem 4.1.1, we have
. 
If there are 
distinct points in the 
-plane with mutually distinct 
coordinates, then there exist a unique polynomial which passes through all given points with degree 
. Let's find the polynomial.
Let 
be 
distinct points in the 
-plane with mutually distinct 
coordinates. We want to find a polynomial of degree 
which passes through all given points. Let
be such a polynomial.
points satisfy the given polynomial, we have
.
are mutually distinct, the coefficient matrix has
.
is called 
. Now we introduce how to compute the determinant of Vandermonde matrix. For the case
,
.
Similarly, as we illustrated in the above case, the determinant of Vandermonde matrix 
with degree 
is product of 
(with 
). That is,
.
■
Reference video: 
Practice site: 
matrix 
and a vector 
, 
is a vector in 
. One of the important questions in applied problems is 
, which makes both 
and 
parallel?”
be an 
matrix. For a nonzero vector 
, if there exist a scalar 
which satisfies 
, then 
is called an 
, and 
is called an 
corresponding to 
.
and 
. Then 
=
. Hence 3 is an eigenvalue of 
, and 
is an eigenvector corresponding to 3. 
, for any 
, the only eigenvalue of identity matrix 
is 
. Also, any nonzero vector 
is an eigenvector of 
corresponding to 1. 
If 
is an eigenvector of 
corresponding to 
, then 
is also an eigenvector of 
corresponding to 
for 
.
.
, the system of linear equations 
should have nonzero solution. Therefore, the 
should hold. 
is called the 
be 
matrix and 
is a scalar, then the following statements are equivalent: 
is an eigenvalue of 
.
is a solution of the characteristic equation 
.
has a nontrivial solution.
.
satisfies 
. Then,
(1)
.
.
Do eigenvalues exist for any square matrix?
roots 
, that is, 
, on the complex plane.
That is, a real square matrix 
with degree 
always has 
eigenvalues in complex domain. However, in this textbook we have limited the scalar as real numbers, and hence there is no eigenvalues means there is no 
.
, solve 
.
, solve 
.
.
, solve 
.
, solve 
.
and 
are real numbers not simultaneously become zero)
For a triangular matrix 
with degree 
, the main diagonal components of 
are 
(
). Therefore, the characteristic polynomial of 
is 
, and hence the eigenvalues of the triangular matrix 
are its main diagonal components, 
.
.
, 
's eigenvalues are 
. 
be an eigenvalue of 
matrix 
. Then the solution space of the system of linear equations 
is called 
corresponding to 
.
That is, an eigenspace of 
corresponding to 
is the set of all eigenvectors of 
corresponding to 
and the zero vector, which is a subspace of 
.
in 
, find eigenspaces of 
corresponding to each eigenvalue 
.
,
, by solving 
, we have
, by solving 
, we have
(
, 
)
of 
even or odd?
(
)
(
)
(
)
(By Theorem 4.1.9)
=-1. 
Let 
be 
matrix and 
, find the followings.
(2) 
(3) 
(4) 
■
For given matrices.
, 
. 
.
,
인 특수한 행렬에 대해서 
를 구해보면,
,
는 행렬 
의 2번째-3번째 열을 바꾸고, 그 다음 2번째-3번째 행을 바꾼 형태이므로 Theorem 4.1.3 에 의해 
이 성립한다.
matrix 일 때에도 성립한다는 것을 보일 때 Determinant 의 성질이 완벽히 참이라는 것이 증명이 되지만 여기서는 그 경우까지 고려하면 약간 복잡해지기 때문에 이 페이지에 싣지는 않을 것이다. 그렇지만 일반적인 matrix 에 대하여 
가 성립하며 이는 upper triangular matrix 또는 diagonal matrix 등 특수한 matrix 의 성질을 이용하여 증명할 수 있다.
and 
, the given matrix is invertible?
invertible => 
.
is a lower triangular matrix. By Theorem 4.1.9,
.
■ 
For given matrices,
.
.
.
Find all cofactors of the following matrices.
(2) 
, 
, 
, 
, 
, 
, 
(Using 
column 
.
.
.
.
Find the adjoint matrix 
of the matrix 
form (Problem 8).
(2) 
by cofactor expansion
by cofactor expansion
= 
, 
=
=
, 
, 
=
=
=
=
=
, 
=
, 
=
(
Cramer's Rule)
=
=
, 
=
=
, 
=
=
, 
, 
=8, 
=4, 
=-8 , 
=0, 
=12
=
, 
=
, 
=
, 
=
=
=
, 
=
=-1, 
=
=0, 
=
=
, 
, 
=-4, 
=-4, 
=-4, 
=-8
=
, 
=
, 
=
=
=1, 
=
=1, 
=
=2 
, 
, 
=
=
=
=
=
=
, 
=
, 
=
, 
=
=
=
, 
=
=
, 
=
=
, 
=
=
Solve the following problems by using the determinant of Vandermonde matrix.
which passes
.
The Vadermonde matrix is 
.
, 
Vardermonde matrix is 
, 
Solve the following problems by using the determinant.
(Area of parallelogram)
(Area of parallelepiped)
Find the eigenvalues and eigenvectors of the following matrices.
(2) 
, 
, 
, then 
(
From 
)
, where 
.
. then 
.
, where 
.
,
, then
, where 
.
, then
, where 
.
, then
, where 
.
, then
, where 
.
에 대하여 
를 만족하는 
, 
이 존재하면
이 되어야 하므로 
.
, 
.
, 
는 위 이차방정식의 해이므로, 
, 
.
Explain why 
for the following matrix 
.
has identical rows.
(
Theorem.4.1.4) ■
Show that for two square matrices 
and 
, if 
for an invertible matrix 
, then 
.
, 
에 대하여 
(New) Simplify the following determinant.
. (One example)
■
임을 알 수 있다. ]
(
From Theorem 4.2.2) ]
# multiply 
# 
# determinant each side
# using 
# using 
# devide 
both side
■
Let 
be a 
matrix, and assume that
.
. Which relation does this value have with 
?
.
, 
]
(Using 
, (
From Problem p4)
, then
,
(Using 
이 성립한다는 것을 보였기 때문에 이를 응용하여 간단히 
를 구할 수 있었다. ]
By using the Cramer's rule,
which passes through the following four points.
,)
, 
, 
■
Let the characteristic polynomial of matrix 
be 
. Find eigenvalues of matrix 
.
,
, 
(multiple root)
, so 
Eigenvalues of matrix 
are 
, 
. ■
차 정사각행렬 
의 고윳값을 
라고 할 때 (단, 
, 
, 
, 
), 
인 자연수 
에 대하여 
의 고윳값이 
임을 보이자.
일 때 
일 때 
임이 성립한다고 가정하면,
일 때
일 때도 성립한다.
의 고윳값은 
이다. ]
Find the eigenspaces of 
, corresponding to each eigenvalue and show that they are orthogonal to each other in the plane.
,
, 
, 
, 
(단, 
,
)에 대해서 내적한 값을 구하였는데 어차피 scalar 값을 고려할 필요가 없기 때문에 eigenvector 들의 내적 값을 구하여 
이 되는지만 확인하면 된다. ]
Find the characteristic polynomial of the following matrix. And find the roots of the polynomial by using the Sage.
is the determinant of 
.
, 
, 
,
, 
