Elementary Linear AlgebraAnton & Rorres, 9th Edition
Lecture Set – 02Chapter 2: Determinants
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Chapter Content
Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants
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2-1 Minor and Cofactor
Definition Let A be mn
The (i,j)-minor of A, denoted Mij is the determinant of the (n-1) (n-1) matrix formed by deleting the ith row and jth column from A
The (i,j)-cofactor of A, denoted Cij, is (-1)i+j Mij
Remark Note that Cij = Mij and the signs (-1)i+j in the definition of
cofactor form a checkerboard pattern:
...
...
...
...
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2-1 Example 1
Let
The minor of entry a11 is
The cofactor of a11 is C11 = (-1)1+1M11 = M11 = 16
Similarly, the minor of entry a32 is
The cofactor of a32 is C32 = (-1)3+2M32 = -M32 = -26
8 4 1
6 5 2
4 1 3
A
168 4
6 5
8 4 1
6 5 2
4 1 3
11
M
266 2
4 3
8 4 1
6 5 2
4 1 3
32
M
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2-1 Cofactor Expansion The definition of a 3×3 determinant in terms of minors and
cofactors det(A) = a11M11 +a12(-M12)+a13M13
= a11C11 +a12C12+a13C13
this method is called cofactor expansion along the first row of A
Example 2
10)11)(1()4(3 4 5
4- 20
2 5
3 2 1
2 4
3 43
2 4 5
3 4 2
0 1 3
)det(
A
2-1 Cofactor Expansion det(A) =a11C11 +a12C12+a13C13 = a11C11 +a21C21+a31C31
=a21C21 +a22C22+a23C23 = a12C12 +a22C22+a32C32
=a31C31 +a32C32+a33C33 = a13C13 +a23C23+a33C33
Theorem 2.1.1 (Expansions by Cofactors) The determinant of an nn 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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2-1 Example 3 & 4 Example 3
cofactor expansion along the first column of A
Example 4 smart choice of row or column
det(A) = ?
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1)3(5)2)(2()4(3 3 4
0 1 5
2 4
0 1 )2(
2 4
3 43
2 4 5
3 4 2
0 1 3
)det(
A
1002
12-01
2213
1-001
A
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2-1 Adjoint of a Matrix If A is any nn matrix and Cij is the cofactor of aij, then the
matrix
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)
Remarks If one multiplies the entries in any row by the corresponding
cofactors from a different row, the sum of these products is always zero.
nnnn
n
n
C CC
C CC
C CC
21
22221
11211
2-1 Example 5 Let
a11C31 + a12C32 + a13C33 = ?
Let
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333231
232221
131211
aaa
aaa
aaa
A
131211
232221
131211
'
aaa
aaa
aaa
A
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2-1 Example 6 & 7
Let
The cofactors of A are: C11 = 12, C12 = 6, C13 = -16, C21 = 4, C22 = 2, C23 = 16, C31 = 12, C32 = -10, C33 = 16
The matrix of cofactor and adjoint of A are
The inverse (see below) is
042
361
123
A
16 10 12
16 2 4
16 6 12
16 16 16
10 2 6
12 4 12
)(adj A
16 16 16
10 2 6
12 4 12
64
1)(adj
)det(
11 AA
A
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Theorem 2.1.2 (Inverse of a Matrix using its Adjoint) If A is an invertible matrix, then
Show first that )(adj
)det(
11 AA
A
)det(2211 ACaCaCa ininiiii
IAAA )det()(adj
)(02211 jiCaCaCa jninjiji
Theorem 2.1.3 If A is an n × n triangular matrix (upper triangular, lower
triangular, or diagonal), then det(A) is the product of the entries on the main diagonal of the matrix; det(A) = a11a22…ann
E.g.
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44434241
333231
2221
11
0
00
000
aaaa
aaa
aa
a
A
?
40000
89000
67600
15730
38372
2-1 Prove Theorem 1.7.1c A triangular matrix is invertible if and only if its diagonal
entries are all nonzero
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2-1 Prove Theorem 1.7.1d The inverse of an invertible lower triangular matrix is
lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular
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Theorem 2.1.4 (Cramer’s Rule) If Ax = b is a system of n linear equations in n unknowns such
that det(I – A) 0 , then the system has a unique solution. This solution is
where Aj is the matrix obtained by replacing the entries in the column of A by the entries in the matrix b = [b1 b2 ··· bn]T
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)det(
)det(,,
)det(
)det(,
)det(
)det( 22
11 A
Ax
A
Ax
A
Ax n
n
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2-1 Example 9 Use Cramer’s rule to solve
Since
Thus,
832
30643
62
321
321
31
xxx
xxx
x x
8 2 1
30 4 3
6 0 1
,
3 8 1
6 30 3
2 6 1
,
3 2 8
6 4 30
2 0 6
,
3 2 1
6 4 3
2 0 1
321 AAAA
11
38
44
152
)det(
)det(,
11
18
44
72
)det(
)det(,
11
10
44
40
)det(
)det( 33
22
11
A
Ax
A
Ax
A
Ax
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Chapter Content
Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants
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Theorems Theorem 2.2.1
Let A be a square matrix If A has a row of zeros or a column of zeros, then det(A)
= 0.
Theorem 2.2.2 Let A be a square matrix det(A) = det(AT)
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Theorem 2.2.3 (Elementary Row Operations) Let A be an nn matrix
If B is the matrix that results when a single row or single column of A is multiplied by a scalar k, than det(B) = k det(A)
If B is the matrix that results when two rows or two columns of A are interchanged, then det(B) = - det(A)
If B is the matrix that results when a multiple of one row of A is added to another row or when a multiple column is added to another column, then det(B) = det(A)
Example 1
333231
232221
131211
k k k
)det(
aaa
aaa
aaa
B
2-2 Example of Theorem 2.2.3
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Theorem 2.2.4 (Elementary Matrices) Let E be an nn elementary matrix
If E results from multiplying a row of In by k, then det(E) = k
If E results from interchanging two rows of In, then det(E) = -1
If E results from adding a multiple of one row of In to another, then det(E) = 1
Example 2
Theorem 2.2.5 (Matrices with Proportional Rows or Columns) If A is a square matrix with two proportional rows or two
proportional column, then det(A) = 0
Example 3
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2-2 Example 4 (Using Row Reduction to Evaluate a Determinant) Evaluate det(A) where
Solution:
162
9 63
510
-
A
.
1 6 2
5 1 0
3 2 1
3
1 6 2
5 1 0
9 6 3
1 6 2
9 6 3
5 1 0
)det(
A
The first and second rows of A are interchanged.
A common factor of 3 from the first row was taken through the determinant sign
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2-2 Example 4 (continue)
.
165)1)(55)(3(
1 0 0
5 1 0
3 2- 1
)55)(3(
55- 0 0
5 1 0
3 2- 1
3
5- 10 0
5 1 0
3 2- 1
3
-2 times the first row was added to the third row.
-10 times the second row was added to the third row
A common factor of -55 from the last row was taken through the determinant sign.
1 6 2
5 1 0
3 2 1
3)det(
A
2-2 Example 5 Using column operation to evaluate a determinant
Compute the determinant of
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5-137
0360
6072
3001
A
2-2 Example 6 Row operations and cofactor expansion
Compute the determinant of
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3573
5142
11-21
62-53
A
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Chapter Content
Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants
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2-3 Basic Properties of Determinant Since a common factor of any row of a matrix can be
moved through the det sign, and since each of the n row in kA has a common factor of k, we obtain
det(kA) = kndet(A)
There is no simple relationship exists between det(A), det(B), and det(A+B) in general.
In particular, we emphasize that det(A+B) is usually not equal to det(A) + det(B).
2-3 Example 1
det(A+B) ≠det(A)+det(B)
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83
34,
31
13,
52
21ABBA
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Theorems 2.3.1 Let A, B, and C be nn 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. Let
2221
1211
aa
aaA
2221
1211
bb
aaB
2-3 Example 2
Using Theorem 2.3.1
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110
3 0 2
5 7 1
det
741
3 0 2
5 7 1
det
)1(71401
3 0 2
5 7 1
det
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Lemma 2.3.2 If B is an nn matrix and E is an nn elementary matrix,
then det(EB) = det(E) det(B)
Remark: If B is an nn matrix and E1, E2, …, Er, are nn elementary
matrices, then
det(E1 E2 · · · Er B) = det(E1) det(E2) · · · det(Er) det(B)
Theorem 2.3.3 (Determinant Test for Invertibility) A square matrix A is invertible if and only if det(A) 0
Example 3
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642
101
321
A
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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)
Example 4
143
172,
85
31,
12
13ABBA
Theorem 2.3.5
If A is invertible, then
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)det(
1)det( 1
AA
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2-4 Linear Systems of the Form Ax = x Many applications of linear algebra are concerned with
systems of n linear equations in n unknowns that are expressed in the form Ax = x, where is a scalar
Such systems are really homogeneous linear in disguise, since the expresses can be rewritten as (I – A)x = 0
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2-3 Eigenvalue and Eigenvector The eigenvalues of an nn matrix A are the number for
which there is a nonzero x 0 with Ax = x.
The eigenvectors of A are the nonzero vectors x 0 for which there is a number with Ax = x.
If Ax = x for x 0, then x is an eigenvector associated with the eigenvalue , and vice versa.
2-3 Eigenvalue and Eigenvector Remark:
A primary problem of linear system (I – A)x = 0 is to determine those values of for which the system has a nontrivial solution.
Theorem (Eigenvalues and Singularity) is an eigenvalue of A if and only if I – A is singular,
which in turn holds if and only if the determinant of I – A equals zero:
det(I – A) = 0 (the so-called characteristic equation of A)
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2-3 Example 5 & 6 The linear system
The characteristic equation of A is
The eigenvalues of A are = -2 and = 5 By definition, x is an eigenvector of A if and only if x is a nontrivial
solution of (I – A)x = 0, i.e.,
If = -2, x = [-t t]T – one eigenvector If = 5, x = [3t/4 t]T – the other eigenvector
221
121
24
3
xxx
xxx
2 4
3 1I
Axx A
0103or 0 2 4
3 1 )det( 2
AI
0
0
2 4
3 1
2
1
x
x
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Theorem 2.3.6 (Equivalent Statements) If A is an nn 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 n1 matrix b Ax = b has exactly one solution for every n1 matrix b det(A) 0
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2-4 Permutation A permutation of the set of integers {1,2,…,n} is an arrangement of
these integers in some order without omission repetition Example 1
There are six different permutations of the set of integers {1,2,3}:(1,2,3), (2,1,3), (3,1,2), (1,3,2), (2,3,1), (3,2,1).
Example 2 List all permutations of the set of integers {1,2,3,4}
1
2 3 4
3 4 2 4 2 3
4 3 4 2 3 2
2
1 3 4
3 4 1 4 1 3
4 3 4 1 3 1
3
1 2 4
2 4 1 4 1 2
4 2 4 1 2 1
4
1 2 3
2 3 1 3 1 2
3 2 3 1 2 1
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2-4 Inversion An inversion is said to occur in a permutation (j1, j2, …,
jn) whenever a larger integer precedes a smaller one.
The total number of inversions occurring in a permutation can be obtained as follows:
Find the number of integers that are less than j1 and that follow j1 in the permutation;
Find the number of integers that are less than j2 and that follow j2 in the permutation;
Continue the process for j1, j2, …, jn. The sum of these number will be the total number of inversions in the permutation
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2-4 Example 3
Determine the number of inversions in the following permutations: (6,1,3,4,5,2) (2,4,1,3) (1,2,3,4)
Solution: The number of inversions is 5 + 0 + 1 + 1 + 1 = 8 The number of inversions is 1 + 2 + 0 = 3 There no inversions in this permutation
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2-4 Classifying Permutations A permutation is called even if the total number of inversions
is an even integer and is called odd if the total inversions is an odd integer
Example 4 The following table classifies the various permutations of {1,2,3}
as even or odd Permutation Number of Inversions
classification
(1,2,3)
(1,3,2)
(2,1,3)
(2,3,1)
(3,1,2)
(3,2,1)
0
1
1
2
2
3
even
odd
odd
even
even
odd
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2-4 Elementary Product By an elementary product from an nn matrix A we shall
mean any product of n entries from A, no two of which come from the same row or same column.
Example The elementary product of the matrix
is
333231
232221
131211
aaa
aaa
aaa
312213312312322311
322113332112332211
aa aaa aaaa
aa aaa aaaa
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2-4 Signed Elementary Product An nn matrix A has n! elementary products. There are
the products of the form a1j1a2j2 ··· anjn
, where (j1, j2, …, jn) is a permutation of the set {1, 2, …, n}.
By a signed elementary product from A we shall mean an elementary a1j1
a2j2 ··· anjn multiplied by +1 or -1.
We use + if (j1, j2, …, jn) is an even permutation and – if (j1, j2, …, jn) is an odd permutation
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2-4 Example 6
List all signed elementary products from the matrices
2221
1211
aa
aa
333231
232221
131211
aaa
aaa
aaa
Elementary
Product
Associated
Permutation
Even or Odd Signed Elementary
Product
a11a22
a12a21
(1,2)
(2,1)
even
odd
a11a22
- a12a21
Elementary
Product
Associated
Permutation
Even or Odd Signed Elementary
Product
a11a22a33
a11a23a32
a12a21a33
a12a23a31
a13a21a32
a13a22a31
(1,2,3)
(1,3,2)
(2,1,3)
(2,3,1)
(3,1,2)
(3,2,1)
even
odd
odd
even
even
odd
a11a22a33
- a11a23a32
- a12a21a33
a12a23a31
a13a21a32
- a13a22a31
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2-4 Determinant Let A be a square matrix. The determinant function is
denoted by det, and we define det(A) to be the sum of all signed elementary products from A. The number det(A) is called the determinant of A
Example 7
211222112221
1211det aaaa aa
aa
322311332112312213322113312312332211
333231
232221
131211
det aaaaaaaaaaaaaaaaaaa
aaa
aaa
aaa
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2-4 Using mnemonic for Determinant The determinant is computed by summing the
products on the rightward arrows and subtracting the products on the leftward arrows
Remark: This method will not work for determinant of 44
matrices or higher!
2221
1211
aa
aa
333231
232221
131211
aaa
aaa
aaa
3231
2221
1211
aa
aa
aa
2-4 Example 8 Evaluate the determinants of
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24
13A
987
654
321
B
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2-4 Notation and Terminology We note that the symbol |A| is an alternative notation for det(A)
The determinant of A is often written symbolically asdet(A) = a1j1
a2j2 ··· anjn
where indicates that the terms are to be summed over all permutations (j1, j2, …, jn) and the + or – is selected in each term according to where the permutation is even or odd
Remark: 44 matrices need 4! = 24 signed elementary products 1010 determinant need 10! = 3628800 signed elementary products! Other methods are required.