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Unformatted text preview: have established the multiplicative property, the
forward part of (*) is cntlCcntlV of the 2x2 case.
Suppose that A is an invertible 3x3 matrix. Let the inverse be A1. Apply the
determinant function to both sides of
1
= 1
1 we obtain det( ∙ ) = 1. Using the multiplicative property,
det( )det(
) = 1.
If det(A) is zero, the left hand side is 0. This yields a contradiction to the
above equation. Therefore, det(A) is nonzero.
Another formula for 3x3 determinant
17. Definitions on permutations. In general, for a positive integer n, a
permutation of 1,2,…,n is represented by a bijective function, say , from
{1,2,…,n} to itself.
1
↓
(1) 2
…
↓
…
(2) … n
↓
( n) There are n! possible bijective functions from {1,2,…,n} to itself. The
collection of all n! of them is denoted by Sn (called the symmetric group in
mathematics). We use the shorthand notation
8 1
(1)
to represent the permutation 2
…
(2) … n
(1) → ( ). For example The permutation 1 → 2, 2 → 3 and 3 → 1 is represented
by
12
23 3
.
1 A transposition is a permutation which exchanges two items, while keeping
the other fixed. For example, the swapping of 1 and 2 is represented by
12
21 3
.
3 The sign (or signature) of the permutation represented by is defined as
sgn( ) = (−1)
where N is the number of transpositions required to transform (1,2,…,n)
to ((1), (2), …, (n)). This is equal to +1 if N is even and 1 if N is odd.
Example for n=3. For the permutation specified by (1)=1, (2)=2, (3)=3, (the identity
permutation), we have N = 0, and hence sgn() = 1.
If is a permutation specified by (1)=2, (2)=1, (3)=3, it is the
exchange of 1 and 2, and sgn() = 1.
If (1)=3, (2)=1, (3)=2, we have N = 2, and sgn() = 1. 18. Using the notation introduced above, we have another formula for 3x3
determinant: For 3x3 matrix A= [aij],
det( ) = sgn( ) () () () ∈ There are six terms in the above summation, each term corresponds to a
bijective function in S3.
nxn determinant in general
19. Given an nxn matrix M = [mij,], we can either define the determinant of M
recursively by Laplace expansion: Expansion on the ith row (i=1,2,…,n).
(−1) + (−1)
9 + ⋯ + (−1) Expansion on the jth column (j=1,2,…,n).
(−1)
+ (−1)
+ ⋯ + (− 1) or by generalizing the formula in (18),
sgn( ) ( )… () () ∈ with the summation running over all n! permutations in Sn. The first
approach is adopted in Kreyszig ’s book, but the second is also common
(see http://en.wikipedia.org/wiki/Determinant). The definition via
expansion has the drawback that we need to check that all expansions
yield the same number. (This is done in Appendix 4 in Kreysizig ’s book.) The
definition in terms of permutations does not have the problem of
welldefinedness, but is less friendly to numerical calculations.
The main result (*) holds for nxn matrix in general, the proof goes along
the same pathway as in the 3x3 case. The properties in (13) are also
satisfied by nxn matrices in general.
20. Practical method for evaluating determinant. Computing an nxn
determinant directly from the two definitions in (19) require n! steps,
which increases exponentially as n increases. A more practical approach is
to apply elementary row operations to transform the matrix to an upper
(or lower) matrix, and calculate the product of the diagonal elements. This
method is based on the properties Subtracting a scalar multiple of a row from another row does not
change the determinant. The determinant of an upper (or lower) triangular matrix is equal to
the product of the diagonal entries.
21. (Caution) When we multiply a matrix by a constant, we multiply all entries
in the matrix. When we multiply a determinant by a constant, we multiply
only the entries in a row or in a column. For constant c,
1
3
1
3 2
=
4
3 2
2
1
=
=
4
34
3 10 2
,
4 2
=
4
3 2
1
=
4
3 2
.
4...
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This document was uploaded on 09/16/2013.
 Spring '13

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