Download section 2.1 and section 2.3

Chapter 2
•
Determinants by Cofactor Expansion
•
Evaluating Determinants by Row Reduction
•
Properties of the Determinants, Cramer’s Rule
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Minor and Cofactor
a b 
Let A be a 2x2 matrix A  

c d 
Then ad-bc is called the determinant of the matrix A, and is denoted by
The symbol det(A).
We will extend the concept of a determinant to square matrices of all orders.
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Minor and Cofactor
Definition
If A is a square matrix, then
• The minor of entry aij, denoted Mij, is defined to be the determinant of
the submatrix that remains after the ith row and jth column are
deleted from A.
• The number (-1)i+j Mij is denoted by Cij and is called the cofactor of
entry aij
Remark
Note that Cij = Mij and the signs
checkerboard pattern:
(1)i  j in the definition of cofactor form a






 
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    ...
    ...
    ...

    ...

   
3
3

Example: Let A   2
 1
1  4
5 6  . Find the minor and cofactor of a12, and a23
4 8 
Solution:
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Cofactor Expansion
Theorem 2.1.1 (Expansions by Cofactors)
The determinant of an nn matrix A can be computed by multiplying
the entries in any row (or column) by their cofactors and adding the
resulting products; that is, for each 1  i, j  n
det(A) = a1jC1j + a2jC2j +… + anjCnj
(cofactor expansion along the jth column)
and
det(A) = ai1Ci1 + ai2Ci2 +… + ainCin
(cofactor expansion along the ith row)
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3
Example: Let A   2
5
1
0
4
3
4 2
. Evaluate det (A) by cofactor expansion
along the first column of A.
Solution:
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Determinant of an Upper Triangular Matrix
Theorem
If A is an nxn triangular matrix (upper triangular, lower triangular, or diagonal),
then det(A) is the product of the entries on the main diagonal of the matrix;
that is, det(A)=a11a22…ann.
Arrow Technique for determinants of 2x2 and 3x3 matrices.
a11
a12
a21 a22
a11 a12
a21 a22
a31 a32
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 a11a22  a12 a21
a13
a23  a11a22 a33  a12 a23a31  a13a21a32  a13a22 a31  a12 a21a33  a11a23a32
a33
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Section 2.3 Properties of Determinants; Cramer’s Rule
Definition (Adjoint of a Matrix)
If A is any nn matrix and Cij is the cofactor of aij, then the matrix
C11 C12  C1n 
C C  C 
2n 
 21 22
 




C
C

C
nn 
 n1 n 2
is called the matrix of cofactors from A. The transpose of this matrix is
called the adjoint of A and is denoted by adj(A)
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 3 2 1 

6
3  . Find the adjoint of A.
Example: Let A   1
 2  4 0 
Solution:
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Theorems
Theorem 2.1.2 (Inverse of a Matrix using its Adjoint)
If A is an invertible matrix, then
A1 
1
adj( A)
det( A)
 3 2 1 

Example: Use theorem 2.1.2. to find the inverse of A   1
6
3


 2  4 0 
Solution:
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Cramer’s Rule
Theorem 2.1.4 (Cramer’s Rule)
If Ax = b is a system of n linear equations in n unknowns such that
det(A)  0 , then the system has a unique solution. This solution is
x1 
det( An )
det( A1 )
det( A2 )
, x2 
,, xn 
det( A)
det( A)
det( A)
where Aj is the matrix obtained by replacing the entries in the jth
column of A by the entries in the matrix
b1 
b 
b   2
 
 
bn 
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Example: Use Cramer’s rule to solve
x1 
 2 x3  6
 3x1  4 x2  6 x3  30
 x1  2 x2  3x3  8
Solution:
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Properties of the Determinant Function
Suppose that A and B are nxn matrices and k is any scalar. We obtain
det(kA)  k n det( A)
But There is no simple relationship exists between det(A), det(B), and det(A+B)
in general. In particular, det(A+B) is usually not equal to det(A) + det(B).
Example: Consider
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1 2 
3 1 
 4 3
A
,
B

,
A

B


1 3
 3 8
2 5




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Properties of the Determinants Sum
Theorem 2.3.1
Let A, B, and C be nn matrices that differ only in a single row, say the r-th, and
assume that the r-th row of C can be obtained by adding corresponding entries in
the r-th rows of A and B. Then
det(C) = det(A) + det(B)
The same result holds for columns.
Example
1
det  2
 1  0
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5 
1
0
3   det  2
 1
4  1 7  (1)
7 5
1
0 3  det  2
 0
4 7 
7
5
0 3
1  1
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Properties of the Determinants Multiplication
Lemma 2.3.2
If B is an nn matrix and E is an nn elementary matrix, then
det(EB) = det(E) det(B)
Theorem 2.3.3 (Determinant Test for Invertibility)
A square matrix A is invertible if and only if det(A)  0
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Theorem 2.3.4
If A and B are square matrices of the same size, then
det(AB) = det(A) det(B)
Theorem 2.3.5
1
If A is invertible, then det( A ) 
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det( A)
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Theorem 2.3.6 (Equivalent Statements)
If A is an nn matrix, then the following are equivalent
•
A is invertible.
•
Ax = 0 has only the trivial solution
•
The reduced row-echelon form of A as In
•
A is expressible as a product of elementary matrices
•
Ax = b is consistent for every n1 matrix b
•
Ax = b has exactly one solution for every n1 matrix b
•
det(A)  0
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