APP. A] VECTORS AND MATRICES 417
(b)∣ ∣ ∣ ∣ ∣ ∣
21 3
46 − 1
51 0∣ ∣ ∣ ∣ ∣ ∣= 2 ( 6 )( 0 )+ 1 (− 1 )( 5 )+ 3 ( 1 )( 4 )− 5 ( 6 )( 3 )− 1 (− 1 )( 2 )− 0 ( 1 )( 4 )= 0 − 5 + 12 − 90 + 2 − 0 = 81General Definition of Determinants
The general definition of a determinant of ordernis as follows:det(A)=∑
sgn(σ )a 1 j 1 a 2 j 2 ...anjnwhere the sum is taken over all permutationsσ={j 1 ,j 2 ,...,jn}of{ 1 , 2 ,...,n}. Here sgn(σ )equals+1or− 1
according as an even or an odd number of interchanges are required to changeσso that its numbers are in the
usual order. We have included the general definition of the determinant function for completeness. The reader
is referred to texts in matrix theory or linear algebra for techniques for computing determinants of order greater
than 3. Permutations are studied in Chapter 5.
An important property of the determinant function is that it is multiplicative. That is:
Theorem A.3: LetAandBbe anyn-square matrices. Then
det(AB)=det(A)·det(B)The proof of the above theorem lies beyond the scope of this text.Determinants and Inverses of 2×2 Matrices
Consider an arbitrary 2×2 matrixA=[
ab
cd]
.Suppose|A|=ad−bc= 0 .Then one can prove thatA−^1 =[
ab
cd]− 1
=[
d/|A|−b/|A|
−c/|A| a/|A|]
=1
|A|[
d −b
−ca]In other words, when|A|= 0 ,the inverse of a 2×2 matrixAis obtained as follows:
(1) Interchange the elements on the main diagonal.(2) Take the negatives of the other elements.(3) Multiply the matrix by 1/|A|or, equivalently, divide each element by|A|.For example, ifA=
[
23
45]
, then|A|=−2 and soA−^1 =1
− 2[
5 − 3
− 42]
=[
−^5232
2 − 1]On the other hand, if|A|= 0 ,thenA−^1 does not exist. Although there is no simple formula for matrices of
higher order, this result does hold in general. Namely:
Theorem A.4:A matrixAis invertible if and only if it has a nonzero determinant.