Chapter 2Chapter 2
DeterminantsDeterminants
With each square matrix it is possible to associate a real
number called the determinant of the matrix.
The value of this number will tell us whether the matrix
is singular.
Case 1: 1×1 Matrices If A=(a) is a 1×1 matrix, then A will have a multiplicative inverse if and only if a≠0. Thus, if we define
det(A)=a
Then A will be nonsingular if and only if det(A) ≠0.
Case 2 : 2×2 Matrices Let
then
2221
1211
aaaa
A
21122211)det( aaaaA
1 The Determinant of A Matrix
Case 3 3×3 Matrices
11 12 13
21 22 23
31 32 33
a a aa a aa a a
322311332112312213322113312312332211)det( aaaaaaaaaaaaaaaaaaA
Definition Let A=(aij) be an nxn matrix and let Mij denote the (n-1)x(n-1) matrix obtained from A by deleting the row and column containing aij. The determinant of Mij is called the minor of aij. We define the cofactor Aij of aij by
( 1)i jij ijA M
Example If , then calculate det(A).
645213452
A
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:
are the cofactors associated with the entries in the first row of A.
11
)det(1112121111
11
nifAaAaAanifa
Ann
njMA jj
j ,,1)det()1( 11
1 where
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.
.,,1,,1
)(det
2211
2211
njandnifor
AaAaAaAaAaAaA
njnjjjjj
ininiiii
Example Evaluate
3102301005400320
Theorem 2.1.2 If A is an nxn matrix, then det(AT)=det(A).
Theorem 2.1.3 If A is an nxn triangular matrix, the determinant of A equals the product of the diagonal elements of A.
Theorem 2.1.4 Let A be an nxn matrix,(1) If A has a row or column consisting entirely of zeros, then det(A)=0.(2) If A has two identical rows or two identical columns, then det(A)=0.
2 Properties of Determinants
Lemma 2.2.1 Let A be an nxn matrix. If Ajk denotes the cofactor of ajk for k=1, … , n, then
(1)
jiifjiifA
AaAaAa jninjiji 0)det(
2211
Effects of row operation on the the value of a determinant
Row Operation I (Two rows are interchanged.)
Suppose that E is an elementary matrix of type I, then
)det()det()det()det( AEAEA
Effects of row operation on the the value of a determinant
Row Operation II (A row of A is multiplied by a nonzero constant.)
Let E denote the elementary matrix of type II formed from I bymultiplying the ith row by the nonzero constant .
)det()det()det()det( AEAEA
Row Operation III (A multiple of one row is added to another row.)
Let E be the elementary matrix of type III formed from I byadding c times the ith row to the jth row.
)det()det()det()det( AEAEA
Effects of row operation on the the value of a determinant
Ⅰ. Interchanging two rows (or columns) of a matrix
changes the sign of the determinant.
Ⅱ. Multiplying a single row or column of a matrix by a
scalar has the effect of multiplying the value of the
determinant by that scalar.
Ⅲ. Adding a multiple of one row (or column) to another
does not change the value of the determinant.
Example Evaluate
436124312
Main Results
Theorem 2.2.2 An n×n matrix A is singular if and only if
det(A)=0
Theorem 2.2.3 If A and B are n×n matrices, then
det(AB)=det(A)det(B)
Definition The Adjoint of a MatrixLet A be an n×n matrix. We define a new matrix called theadjoint of A by
3 Cramer’s Rule
0)det()det(
11 AwhenAadjA
A
nnnn
n
n
AAA
AAAAAA
adjA
21
22212
12111
Example Let
321223212
A
Compute adj A and A-1.
Theorem 2.3.1 (Cramer’s Rule) Let A be an
nxn nonsingular matrix, and let b∈Rn. Let Ai be
the matrix obtained by replacing the ith column
of A by b. If x is the unique solution to Ax=b, th
en
niforAAx i
i ,,2,1)det()det(
Example Use Cramer’s rule to solve
932622
52
321
321
321
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