Determinants - Mathematics
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The document discusses determinants and their properties. It defines determinants as scalars associated with square matrices. Determinants can be computed for matrices of any size using expansions by cofactors along rows or columns. The key properties of determinants are that they are unchanged by row operations and that the inverse of a non-singular matrix can be computed using the adjugate matrix divided by the determinant. The document provides examples of computing the determinant, adjugate, and inverse of matrices.
![Introduction
Every square matrix has associated with it a scalar called its
determinant.
Given a matrix A, we use det(A) or |A| to designate its determinant.
We can also designate the determinant of matrix A by replacing the
brackets by vertical straight lines. For example,
A=[2 1
0 3 ] det(A)=∣
2 1
0 3
∣
Definition 1: The determinant of a 1×1 matrix [a] is the scalar a.
Definition 2: The determinant of a 2×2 matrix is the scalar ad-bc.
For higher order matrices, we will use a recursive procedure to compute
determinants.
[a b
c d]](https://image.slidesharecdn.com/determinants-160621103059/85/Determinants-Mathematics-2-320.jpg)
![Expansion by Cofactors
Definition 1: Given a matrix A, a minor is the determinant of
square submatrix of A.
Definition 2: Given a matrix A=[aij] , the cofactor of the element
aij is a scalar obtained by multiplying together the term (-1)i+j
and the minor obtained from A by removing the ith row and the
jth column.
In other words, the cofactor Cij is given by Cij = (−1)i+j
Mij.
For example,
A=
[
a11 a12 a13
a21 a22 a23
a31 a32 a33
]
M21=∣
a12 a13
a32 a33
∣
M22=∣
a11 a13
a31 a33
∣
⇒C21=(−1)2+1
M 21=−M21
⇒C22=(−1)
2+2
M22=M22](https://image.slidesharecdn.com/determinants-160621103059/85/Determinants-Mathematics-3-320.jpg)
![
Find the adjugate of
Solution:
The cofactor matrix of A:
A=
[−1 3 2
0 −2 1
1 0 −2 ]
[
∣−2 1
0 −2
∣ −∣0 1
1 −2
∣ ∣0 −2
1 0
∣
−∣
3 2
0 −2
∣ ∣
−1 2
1 −2
∣ −∣
−1 3
1 0
∣
∣
3 2
−2 1
∣ −∣
−1 2
0 1
∣ ∣
−1 3
0 −2
∣] =
[4 1 2
6 0 3
7 1 2 ]
Aa
=
[4 6 7
1 0 1
2 3 2 ]
Example of finding adjugate](https://image.slidesharecdn.com/determinants-160621103059/85/Determinants-Mathematics-9-320.jpg)
![Inversion using determinants
Theorem 2: A Aa
= Aa
A = |A| I .
If |A| ≠ 0 then from Theorem 2, A(Aa
∣A∣)=(Aa
∣A∣)A=I
A
−1
=
1
∣A∣
A
a
if ∣A∣≠0
That is, if |A| ≠ 0, then A-1
may be obtained by dividing the
adjugate of A by the determinant of A.
For example, if
then
A=[a b
c d ],
A−1
=
1
∣A∣
Aa
=
1
ad−bc [d −b
−c a ]](https://image.slidesharecdn.com/determinants-160621103059/85/Determinants-Mathematics-10-320.jpg)
![Use the adjugate of to find A-1A=
[−1 3 2
0 −2 1
1 0 −2 ]
Aa
=
[4 6 7
1 0 1
2 3 2 ]
∣A∣=(−1)(−2)(−2)+(3)(1)(1)−(1)(−2)(2)=3
A
−1
=
1
∣A∣
A
a
=
1
3 [
4 6 7
1 0 1
2 3 2 ]=
[
4
3
2 7
3
1
3
0
1
3
2
3
1 2
3
]
Inversion using determinants: example](https://image.slidesharecdn.com/determinants-160621103059/85/Determinants-Mathematics-11-320.jpg)
Determinants - Mathematics
- 1. Determinants
- 2. Introduction Every square matrix has associated with it a scalar called its determinant. Given a matrix A, we use det(A) or |A| to designate its determinant. We can also designate the determinant of matrix A by replacing the brackets by vertical straight lines. For example, A=[2 1 0 3 ] det(A)=∣ 2 1 0 3 ∣ Definition 1: The determinant of a 1×1 matrix [a] is the scalar a. Definition 2: The determinant of a 2×2 matrix is the scalar ad-bc. For higher order matrices, we will use a recursive procedure to compute determinants. [a b c d]
- 3. Expansion by Cofactors Definition 1: Given a matrix A, a minor is the determinant of square submatrix of A. Definition 2: Given a matrix A=[aij] , the cofactor of the element aij is a scalar obtained by multiplying together the term (-1)i+j and the minor obtained from A by removing the ith row and the jth column. In other words, the cofactor Cij is given by Cij = (−1)i+j Mij. For example, A= [ a11 a12 a13 a21 a22 a23 a31 a32 a33 ] M21=∣ a12 a13 a32 a33 ∣ M22=∣ a11 a13 a31 a33 ∣ ⇒C21=(−1)2+1 M 21=−M21 ⇒C22=(−1) 2+2 M22=M22
- 4. Expansion by Cofactors To find the determinant of a matrix A of arbitrary order, a)Pick any one row or any one column of the matrix; b)For each element in the row or column chosen, find its cofactor; c)Multiply each element in the row or column chosen by its cofactor and sum the results. This sum is the determinant of the matrix. In other words, the determinant of A is given by det( A)=∣A∣=∑ j=1 n aij Cij=ai1 Ci1+ai2 Ci2+⋯+ain Cin det ( A )=∣A∣=∑ i=1 n aij Cij=a1j C1j+a2j C2j +⋯+anj Cnj ith row expansion jth column expansion
- 5. Example 1: We can compute the determinant 1 2 3 4 5 6 7 8 9 =T by expanding along the first row, ( ) ( ) ( ) 1 1 1 2 1 35 6 4 6 4 5 1 2 3 8 9 7 9 7 8 + + + = × − + × − + × −T 3 12 9 0= − + − = Or expand down the second column: ( ) ( ) ( ) 1 2 2 2 3 24 6 1 3 1 3 2 5 8 7 9 7 9 4 6 + + + = × − + × − + × −T 12 60 48 0= − + = Example 2: (using a row or column with many zeroes) ( ) 2 3 1 5 0 1 5 2 1 1 1 3 1 3 1 0 + = × − − − 16=
- 6. Properties of determinants Property 1: If one row of a matrix consists entirely of zeros, then the determinant is zero. Property 2: If two rows of a matrix are interchanged, the determinant changes sign. Property 3: If two rows of a matrix are identical, the determinant is zero. Property 4: If the matrix B is obtained from the matrix A by multiplying every element in one row of A by the scalar λ, then | B|= λ|A|. Property 5: For an n × n matrix A and any scalar λ, det(λA)= λn det(A).
- 7. Properties of determinants Property 6: If a matrix B is obtained from a matrix A by adding to one row of A, a scalar times another row of A, then |A|=|B|. Property 7: det(A) = det(AT ). Property 8: The determinant of a triangular matrix, either upper or lower, is the product of the elements on the main diagonal. Property 9: If A and B are of the same order, then det(AB)=det(A) det(B).
- 8. Inversion Theorem 1: A square matrix has an inverse if and only if its determinant is not zero. Below we develop a method to calculate the inverse of nonsingular matrices using determinants. Definition 1: The cofactor matrix associated with an n × n matrix A is an n × n matrix Ac obtained from A by replacing each element of A by its cofactor. Definition 2: The adjugate of an n × n matrix A is the transpose of the cofactor matrix of A: Aa = (Ac )T
- 9. Find the adjugate of Solution: The cofactor matrix of A: A= [−1 3 2 0 −2 1 1 0 −2 ] [ ∣−2 1 0 −2 ∣ −∣0 1 1 −2 ∣ ∣0 −2 1 0 ∣ −∣ 3 2 0 −2 ∣ ∣ −1 2 1 −2 ∣ −∣ −1 3 1 0 ∣ ∣ 3 2 −2 1 ∣ −∣ −1 2 0 1 ∣ ∣ −1 3 0 −2 ∣] = [4 1 2 6 0 3 7 1 2 ] Aa = [4 6 7 1 0 1 2 3 2 ] Example of finding adjugate
- 10. Inversion using determinants Theorem 2: A Aa = Aa A = |A| I . If |A| ≠ 0 then from Theorem 2, A(Aa ∣A∣)=(Aa ∣A∣)A=I A −1 = 1 ∣A∣ A a if ∣A∣≠0 That is, if |A| ≠ 0, then A-1 may be obtained by dividing the adjugate of A by the determinant of A. For example, if then A=[a b c d ], A−1 = 1 ∣A∣ Aa = 1 ad−bc [d −b −c a ]
- 11. Use the adjugate of to find A-1A= [−1 3 2 0 −2 1 1 0 −2 ] Aa = [4 6 7 1 0 1 2 3 2 ] ∣A∣=(−1)(−2)(−2)+(3)(1)(1)−(1)(−2)(2)=3 A −1 = 1 ∣A∣ A a = 1 3 [ 4 6 7 1 0 1 2 3 2 ]= [ 4 3 2 7 3 1 3 0 1 3 2 3 1 2 3 ] Inversion using determinants: example