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The Determinant Function– The 22 matrix
is invertible if ad-bc0. The expression ad-bc occurs so frequently that it has a name; it is called the determinant of the matrix A and is denoted by the symbol det(A).
–We want to obtain the analogous formula to square matrices of higher order.
dc
baA
Cofactor Expansion: Cramer's Rule
• Minors and Cofactors– det(A)=a11a22a33+a12a23a31+a13a21a32-a13a22a31-a12a21a33-a11a23a32 =a11(a22a33-a23a32)-a12(a21a33-a23a31)+a13(a21a32-a22a31)
=a11M11-a12M12+a13M13 where
– Definition: If A is a square matrix, then the minor of entry aij is denoted by Mij and 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+jMij is denoted by Cij and is called the cofactor of entry aij.
• Example: Finding Minors and Cofactors
3231
222113
3331
232112
3332
232211 , ,
aa
aaM
aa
aaM
aa
aaM
841
652
413
A
– The cofactor and minor of an element differ only in sign. To determine the sign relating Cij and Mij, we may use the checkerboard array
• Cofactor Expansions– det(A)=a11M11-a12M12+a13M13=a11C11+a12C12+a13C13
– This method of evaluating det(A) is called cofactor expansion along the first row of A.• Example: Evaluate det(A) by cofactor expansion
along the first row of A.
Cofactor Expansion: Cramer's Rule
245
342
013
A
– Theorem: 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 iin and 1jn,
• Example: Cofactor expansion along the first column– Cofactor expansion and row or column operations can
sometimes be used in combination to provide an effective method for evaluating determinant.• Example: Evaluate det(A) where
Cofactor Expansion: Cramer's Rule
ininiiii
njnjjjjj
CaCaCaA
CaCaCaA
2211
2211
)det(
)det(
3573
5142
1121
6253
A
• Adjoint of a Matrix– If we multiply the entries in any row by the corresponding
cofactors from a different row, the sum of these products is always zero. (This result also holds for columns)• Example: Find a11C31+a12C32+a13C33 for
– Definition: If A is an 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 is denoted by adj(A).
Cofactor Expansion: Cramer's Rule
333231
232221
131211
aaa
aaa
aaa
A
nnnn
n
n
CCC
CCC
CCC
21
22221
11211
– Example: Find the adjoint of A where
– Theorem: If A is an invertible matrix, then
• Example: Find the inverse of A in last example.
• Cramer's Rule– be of marginal interest for computational
purposes, but is useful for studying the mathematical properties of a solution without the need for solving the system
Cofactor Expansion: Cramer's Rule
042
361
123
A
)adj()det(
11 AA
A
– Theorem: 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 solutions is
where Aj is the matrix obtained by replacing the entries in the jth column of A by the entries in the matrix
Example: Use Cramer's rule to solve
Cofactor Expansion: Cramer's Rule
AA
xA
Ax
A
Ax n
n det
det,,
det
det ,
det
det 22
11
nb
b
b
2
1
b
832
30643
62
321
321
31
xxx
xxx
xx
Cramer’s Rule• Solution
• Therefore,
321
643
201
A
328
6430
206
1A
381
6303
201
2A
821
3043
601
1A
11
10
44
40
)det(
)det( 11
A
Ax
11
18
44
72
)det(
)det( 22
A
Ax
11
38
44
152
)det(
)det( 33
A
Ax
Evaluating Determinants by Row Reduction• A Basic Theorem
– Theorem: Let A be a square matrix.
(a) If A has a row of zeros or a column of zeros, then det(A)=0.
(b) det(A) = det(AT).• It is evident from (b) that every theorem about
determinants that contains the word “row” is also true when the word “column” is substituted.
• Triangular Matrices– Theorem: If A is an nn 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.
• Example: Find the determinant of
40000
89000
67600
15730
38372
• Elementary Row Operations– Theorem: Let A be an nn matrix.
(a) If B is the matrix that results when a single row or single column of A is multiplied by a scalar k, then det(B)=k det(A).
(b) If B is the matrix that results when two rows or two columns of A are interchanged, then det(B) = -det(A).
(c) If B is the matrix that results when a multiple of one row of A is added to another row or when a multiple of one column is added to another column, then det(B) = det(A).
• Elementary Matrices– Theorem: Let E be an nn elementary matrix.
(a) If E results from multiplying a row of In by k, then det(E) = k.
(b) If E results from interchanging two rows of In, then det(E) = -1.
(c) If E results from adding a multiple of one row of In to another, then det(E)=1.
Evaluating Determinants by Row Reduction
• Matrices with Proportional Rows or Columns– Theorem: If A is a square matrix with two
proportional rows or two proportional columns, then det(A) = 0.
• Evaluating Determinants by Row Reduction– Idea: to reduce the given matrix to upper
triangular form by elementary row operations, then compute the determinant of the upper triangular matrix, then relate that determinant to that of the original matrix
– Example: Evaluate det(A) where
– The method is well suited for computer evaluation since it is systematic and easily programmed.
Evaluating Determinants by Row Reduction
162
963
510
A
Properties of the Determinant Function• Basic Properties of Determinants
– Let A and B be nn matrices and k be any scalar. We have det(kA)=kndet(A).
– det(A+B) is usually not equal to det(A)+det(B).
• Example:
• Interesting example:
– Theorem: Let A, B, and C be nn matrices that differ only in single row, say the rth, and assume that the rth row of C can be obtained by adding corresponding entries in the rth rows of A and B. Then det(C) = det(A) + det(B). The same result holds for columns.
31
13 ,
52
21BA
2221
1211
2221
1211 and bb
aaB
aa
aaA
• Determinant of a Matrix Product– If A and B are square matrices of the same size, then
det(AB)=det(A)det(B)– Lemma: If B is an nn matrix and E is an nn elementary
matrix, then det(EB)=det(E)det(B).
• Det(E1E2...ErB)=det(E1)det(E2)...det(Er)det(B)
• Determinant Test for Invertibility– Theorem: A square matrix A is invertible if and only if
det(A)0.– A square matrix with two proportional rows or columns
is not invertible.– Theorem: If A and B are square matrices of the same
size, then det(AB)=det(A)det(B).
Properties of the Determinant Function
– Example: Verifying det(AB)=det(A)det(B)
– Theorem: If A is invertible, then
Properties of the Determinant Function
85
31 ,
12
13BA
)det(
1)det( 1
AA
• Summary– Theorem: If A is an nn matrix, then the
following are equivalent.(a)A is invertible.
(b)Ax=0 has only the trivial solution.
(c)The reduced row-echelon form of A is In.
(d)A is expressible as a product of elementary matrices.
(e)Ax=b is consistent for every n1 matrix b.
(f) Ax=b has exactly one solution for every n1 matrix b.
(g)det(A)0.
Properties of the Determinant Function
• Definition of a Determinant– 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.
The Determinant Function
• Example: List all elementary products from the matrices
– An nn matrix A has n! elementary products of the form
– signed elementary product from A: an elementary product multiplied by +1 or –1. We use + if (j1,j2,...,jn) is an even permutation and the – if (j1,j2,...,jn) is an odd permutation.
• Example: List all signed elementary products from the matrices
The Determinant Function
333231
232221
131211
2221
1211 )( )(
aaa
aaa
aaa
baa
aaa
nnjjj aaa 21 21
333231
232221
131211
2221
1211 )( )(
aaa
aaa
aaa
baa
aaa
• Definition: 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: Find the determinants of matrices
– mnemonic method(only for 22 and 33 matrices)
• Example: Evaluate the determinants of
The Determinant Function
333231
232221
131211
2221
1211 )( )(
aaa
aaa
aaa
baa
aaa
3231
2221
1211
333231
232221
131211
2221
1211
aa
aa
aa
aaa
aaa
aaa
aa
aa
987
654
321
and 24
13BA
• Notation and Terminology– The symbol |A| is an alternative notation for
det(A).– Although the determinant of a matrix is a
number, it is common to use the term “determinant” to refer to the matrix whose determinant is being computed.
– The determinant of A is often written symbolically as
– Evaluating determinants directly from the definition leads to computational difficulties.
The Determinant Function
nnjjj aaaA
21 21)det(