Download Chapter 2 Determinants

Chapter 2---Section 1
Chapter 2
The Determinants of a Matrix
Determinants
With each square matrix it is possible to associate a real number
called the determinant of the matrix. The value of this number tells us
whether the matrix is nonsingular.
§1. The Determinant of a Matrix
New words and phrases
Determinant 行列式
Minor 余子式
Cofactor 代数余子式
1.1 Introduction and Definition
First we consider three simple cases. Notice that A is nonsingular if and
only if A is row equivalent to I.
Case 1 1x1 Matrices
If A=(a) , we define det(A)=a. A is nonsingular if and only if det(A) is
not zero.
Case 2 2x2 Matrices
If we use row operations, we see that A is nonsingular if and only if
a11a22  a12 a21  0 . We define
Chapter 2---Section 1
The Determinants of a Matrix
det(A)= a11a22  a12 a21 .
We can use the determinants of 1x1 matrices to express det(A)
Notation We can refer to the determinant of a specific matrix by
enclosing the array between vertical lines.
a11
a12
a21 a22
 a11a22  a12 a21
Case 3 3x3 Matrices
This case is left to students. In this case, if we define that
det(A)=  a11a22 a33  a12a23a31  a13a21a32  a13a22a31  a12a21a33  a11a23a32
Then we see that A is nonsingular if and only if det(A) is not zero.
a11
a12
a13
a21
a22
a23
a31
a32
a33
We can use the determinants of 2x2 matrices to express det(A).
★Definition Let A   aij  be an nxn matrix and let M ij denote the
(n-1)x(n-1) matrix obtained from A by deleting the row and column
containing aij . The determinant of M ij is called the minor of aij . We
define the cofactor Aij of aij by
Aij  (1)i  j det(M ij )
Chapter 2---Section 1
The Determinants of a Matrix
★Definition The determinant of an nxn matrix A, denoted det(A), is
a scalar associated with the matrix A that is defined inductively as
follows:
if n  1
a11
det( A)  
a11 A11  a21 A21 
 an1 An1
if n  1
(expansion along the first column of A)
where Aj1  (1)1 j det(M j1 ) ， j=1, 2, …, n are the cofactors associated with
the entries in the first column of A.
1.2 Determinants of Triangular matrices
Theorem Let A be a lower triangular matrix (or, upper triangular matrix).
Then, det(A) is the product of the diagonal elements of A.
Proof
If A is upper triangular, then it is easy to prove by induction.
If A is lower triangular, then
det( A)  a11 A11  a21 A21 
 an1 An1 =
n
a
i 1
i1
Ai1
It is sufficient to show that Aj1  0 for j>1. Notice that A1 j is still upper
triangular and has one row of all zeros, and the diagonal contains at least
one element of zero. By induction, the statement is true.
*

*
*

*
0 0 0

* 0 0
* * 0

* * *
Det(I)=1. If A is triangular, then det( AT )  det( A)
Chapter 2---Section 2 Properties of Determinants
§2. Properties of Determinants
New words and phrases
Property 性质
Swap 交换
Induction 归纳法
Invertibility 可逆性
2.1 Effects of Row Operations on the Values of the
Determinants
(1) Row Operations of Type I:
Theorem Suppose that E is an elementary matrix of type I, Then
det(EA)=det(E)det(A)=-det(A),
det( ET )  det( E ) =-1,
det( E 1 )  1/ det( E )
Proof
Step 1: Prove the theorem under the assumption that column 1 is
swapped with column k, for k>1. If this is proved, then swapping column
k and column k’ will be equivalent to performing three swaps: first
swapping column 1 and column k, then swapping column 1 and column
k’, and finally swapping column 1 and column k.
The proof is by induction on n. The base case n=1 is completely
trivial.( Or, if you prefer, you may take n=2 to be the base case, and the
Chapter 2---Section 2 Properties of Determinants
theorem is easily proven using the formula for the determinant of a 2x2
matrix.) Suppose that the statement is true for (n-1)x(n-1) matrices.
det( A)  a11 A11 
 ak1 A21 
 an1 An1 =
det( B)  b11 B11 
 bk1 Bk1 
 bn1Bn1 =
 ak 1

 a21


B=EA=  a( k 1)1
 a11


 a
 n1
 a11

 a21


A=  a( k 1)1
 ak 1


 a
 n1
ak 2
ak 3
a22
a23
a( k 1)2
a( k 1)3
a12
a13
an 2
an 3
a12
a13
a22
a23
a( k 1)2
a( k 1)3
ak 2
ak 3
an 2
an 3
n
a
i 1
i1
Ai1
n
b B
i 1
i1
i1
akn 

a2 n 


a( k 1) n 
a1n 


ann 
a1n 

a2 n 


a( k 1) n 
akn 


ann 
If i 1, k , then Bi1   Ai1 because M iA1 and M iB1 are the same
except for two rows being swapped （交换）and bi1  ai1 .
For i=1 and i=k
b11  ak1 , B11  (1)11 det( M11B )  det( M11B )  ( 1) k 2 det( M kA1)  ( 1) k det( M kA1)
(total of k-2 swaps)
b11 B11  ak1 (1) k det( M kA1 )  ak1 (1) k 1 det( M kA1 )  ak1 Ak1 ,
bk1  a11
bk1 Bk1  a11 (1) k 1 det( M kB1 )  a11 (1) k 1 (1) k 2 det( M 11A )  a11 det( M 11A )   a11 A11
Chapter 2---Section 2 Properties of Determinants
Corollary If two rows of A are equal, then det(A)=0
(2) Row Operations of Type II
Theorem
If E is an elementary matrix of type II, then
det(EA)=det(E)det(A)=  det(A),
det( E T )  det( E )  
det( E 1 )  1/ det( E ) ,
Proof Use induction （归纳法）on n. If B=EA, then M iB1  aM iA1
for k is not equal to k.
bi1 Bi1  bi1 (1)i 1 det( M iB1 )  ai1 (1)i 1 det( M iA1 )   ai1 Ai1
For i=k, bk1Bk1   ai1 (1)i 1 det( M kB1 )   ai1 (1)i 1 det( M kA1 )   ai1 Ak1
Corollary If A has a row of all zeros, then det(A)=0
(3) Row Operations of Type III
Theorem Suppose that
Chapter 2---Section 2 Properties of Determinants
 a11


 ai1

C= 
 ak1  a 'k1


 a
n1

a12
a13
ai 2
ai 3
ak 2  a 'k 2
a k 3  a 'k 3
an 2
an 3
 a11


 ai1

A= 
 ak 1


a
 n1
 a11


 ai1

B= 
 a 'k 1


a
 n1



ain 


akn  a 'kn 


ann 
a1n
a12
a13
ai 2
ai 3
ak 2
ak 3
an 2
an 3
a12
a13
ai 2
ai 3
a 'k 2
a 'k 3
an 2
an 3
a1n 


ain 


akn 


ann 
a1n 


ain 


a 'kn 


ann 
Then det(C)=det(A)+det(B)
Proof: Use induction to show that
ci1Ci1  ai1 Ai1  bi1Bi1 for i=1, 2, …, n.
Theorem If E is an elementary matrix of type III, then
det(EA)=det(E)det(A)=det(A),
det( E )  det( E T )  1 ,
det( E 1 )  1/ det( E )
Chapter 2---Section 2 Properties of Determinants
Corollary If two rows of A are proportional, then det(A)=0.
2.2 Determinants and invertibility
Theorem A is nonsingular if and only if det(A) is not zero.
Proof A  Ek
E2 E1U . Use induction to show that
det( A)  det( Ek )
det( E2 ) det( E1 ) det(U ) , where U is upper triangular.
det(A) is zero if and only if its reduced echelon form U has a
determinant of zero. If A is singular, then U must have a row of all zeros
and hence det(U)=0(U is also triangular). If A is nonsingular, then U=I,
det(A) is not zero.
2.3 Determinants of Products of Matrices
Theorem det(AB)=det(A)det(B)
Proof: If det(B)=0, then B is singular, then AB is also singular. So
both sides are equal.
If det(A)=0 and det(B) is not zero, then AY=0 has a nontrivial
solution Y. ABX=0 has a nontrivial solution X  B1Y , so AB is singular.
det(AB)=0.
If det(A) is not zero, the A is row equivalent to I. A  Ek
E2 E1 .
Chapter 2---Section 2 Properties of Determinants
Then by the result above, the theorem is proven.
2.4 Determinants of Transposes
Theorem det( A)  det( AT )
Proof Express A in row echelon form U.
A  Ek
E2 E1U , where U is upper triangular and E’s are elementary
matrices.
AT  U T E1T E2T
Ek T
For triangular matrices and elementary matrices B, we have
det( B)  det( BT )
2.5 Effects of Column Operations on the Values of the Determinants
Since det(AE)=det(A)det(E), we have
(1) Interchanging two columns of a matrix changes the sign of the
determinant.
(2) Multiplying a single column of a matrix by a scalar has the
effect of multiplying the value of the determinant by that scalar.
(3) Adding a multiple of one column to another does not change
the value of the determinant.
Chapter 2---Section 2 Properties of Determinants
2.6 Cofactor expansion along any column or row
Theorem 2.1.1 If A is an nxn matrix with n  2 , then det(A) can be
expressed as a cofactor expansion using any row or column of A.
Proof: Let
 a11


 ai1

A= 
 ak 1


a
 n1
 a1 j


 aij

B= 
 akj


a
 nj
a12
a13
a1 j
ai 2
ai 3
aij
ak 2
ak 3
akj
an 2
an 3
anj
a12
a13
a11
ai 2
ai 3
ai1
ak 2
ak 3
ak 1
an 2
an 3
an1
a1n 


ain 


akn 


ann 
a1n 


ain 


akn 


ann 
Then det(A)=-det(B)
n
det( B)   bi1 (1)1i det( M iB1 )
i 1
n
  aij (1)1i det( M iB1 )
(compare
i 1
between M iB1 and M ijA )
n
  aij (1)1i (1) j  2 det( M ijA )
i 1
n
n
i 1
i 1
  aij (1)i  j det( M ijA )   aij Aij
the
difference
Chapter 2---Section 2 Properties of Determinants
n
Hence, det( A)   det( B)   aij Aij .
i 1
This proves that det(A) can be expanded along any column of A. Since
det( A)  det( AT ) , det(A) can be expanded along any row of A.
Assignment
Hand in: 11, 12, 15, 16, 17,
Not required: 19, 20,
Chapter 2---Section 2 Properties of Determinants
§3. Cramer’s Rule
New words and phrase
Cramer’s rule 克莱姆法则
Adjoint matrix 伴随矩阵
Expand 展开
In this section, we learn a method for computing the inverse of a
nonsingular matrix A using determinants. We also learn a method for
solving AX=B using determinants. Both methods are dependent on the
following lemma.
Lemma 2.2.1 Let A be an nxn matrix. If A jk denotes the cofactor of
a jk , then
n
a
k 1
ik
Ajk  det( A) ij
n
and
a
k 1
ki
Akj  det( A) ij
Proof If i=j, then it is obvious. If i is not equal to j, then consider a matrix
obtained by replacing the jth row of A by the ith row of A and expand
along the jth row.
Chapter 2---Section 2 Properties of Determinants
 a11


 ai1

B= 
 ai1


a
 n1
a12
a13
a1k
ai 2
ai 3
aik
ai 2
ai 3
aik
an 2
an 3
ank
a1n 


ain 

 , det(B)=0
ain 


ann 
Let A be an nxn matrix, we define a new matrix called the adjoint of
A
 A11


 Ai1

adjA= 
 Ak1


A
 n1
A12
A13
A1 j
Ai 2
Ai 3
Aij
Ak 2
Ak 3
Akj
An 2
An 3
Anj
T
A1n   A11
 
 
Ain  
 
 =
Akn   A1 j
 
 
Ann   A1n
A21
A31
A2 j
A3 j
A2 n
A3n
An1 





Anj 


Ann 
To form the adjoint, we must replace each term by its cofactor and
then transpose the resulting matrix.
By the lemma above, we obtain
A(adjA)=(adjA)A=det(A)I
If A is nonsingular, det(A) is not zero, thus, A1 
1
adjA
det( A)
Theorem 2.3.1 Let A be an nxn nonsingular matrix, an let b  Rn . Let A j
be the matrix obtained by replacing the jth column of A by b. If x is the
unique solution to AX=b, then
xj 
det( Aj )
det( A)
Chapter 2---Section 2 Properties of Determinants
Cramer’s rule gives a convenient method for expressing the solution
to an nxn system of linear equations in terms of determinants.
Proof
X  A1B 
 a11


 ai1

Aj  
 ak1


a
 n1
1
(adjA) B
det( A)
a12
a13
b1
ai 2
ai 3
bi
ak 2
ak 3
bk
an 2
an 3
bn
a1n 


ain  n

   bs Asj   A1 j
s 1
akn 


ann 
A2 j
A Summary on determinants
 Definition of determinants
 Determinants of triangular matrices
 Effects of row operations on determinants
 Determinants and Nonsingularity
 Determinants of products of matrices
 Determinants of transposes
 Effects of column operations on the values of determinants
 Expansion along any row or column
 Cramer’s rule
 Another way to calculate the inverse of a matrix
 b1 
 
b
Anj   2 
 
 
 bn 
Chapter 2---Section 2 Properties of Determinants
Assignment
Hand in: 6, 7, 8, 10, 11, 12, 13
Not required: 14