416 VECTORS AND MATRICES [APP. A
Then
AB=[
6 − 5 − 10 + 10
3 − 3 − 5 + 6]
=[
10
01]
and BA=[
6 − 515 − 15
− 2 + 2 − 5 + 6]
=[
10
01]ThusAandBare inverses.
It is known thatAB=Iif and only ifBA=I; hence it is only necessary to test one product to determine
whether two matrices are inverses. For example,
⎡
⎣102
2 − 13
418⎤
⎦⎡
⎣− 1122
− 401
6 − 1 − 1⎤
⎦=⎡
⎣− 11 + 0 +12 2+ 0 − 22 + 0 − 2
− 22 + 4 + 18 4 + 0 − 34 − 1 − 3
− 44 − 4 +48 8+ 0 − 88 + 1 − 8⎤
⎦=⎡
⎣100
010
001⎤
⎦Thus the two matrices are invertible and are inverses of each other.
A.9Determinants
To eachn-square matrixA=[aij]we assign a specific number called thedeterminantofAand denoted by
det(A)or|A|or ∣
∣ ∣ ∣ ∣ ∣ ∣ ∣
a 11 a 12 ··· a 1 n
a 21 a 22 ··· a 2 n
..............................
an 1 an 2 ··· ann
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣We emphasize that a square array of numbers enclosed by straight lines, called adeterminant of order n,isnot
a matrix but denotes the number that the determinant function assigns to the enclosed array of numbers, i.e., the
enclosed square matrix.
The determinants of order 1, 2, and 3 are defined as follows:
∣
∣a 11
∣
∣=a 11∣
∣
∣
∣a 11 a 12
a 21 a 22∣
∣
∣
∣=a^11 a^22 −a^12 a^21
∣ ∣ ∣ ∣ ∣ ∣
a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33∣ ∣ ∣ ∣ ∣ ∣=a 11 a 22 a 33 +a 12 a 23 a 31 +a 13 a 21 a 32 −a 13 a 22 a 31 −a 12 a 21 a 33 −a 11 a 23 a 32The diagram in Fig. A-3(a) may help the reader remember the determinant of order 2. That is, the determinant
equals the product of the elements along the plus-labeled arrow minus the product of the elements along the
minus-labeled arrow. There is an analogous diagram to remember a determinant of order 3 which appears in
Fig. A-3(b). For notational convenience, we have separated the three plus-labeled arrows and the three minus-
labeled arrows. We emphasize that there are no such diagrammatic tricks to remember determinants of higher
order.
Fig. A-3EXAMPLE A.8
(a)∣
∣
∣
∣54
23∣
∣
∣
∣=^5 (^3 )−^4 (^2 )=^15 −^8 =^7 ,∣
∣
∣
∣21
− 46∣
∣
∣
∣=^2 (^6 )−^1 (−^4 )=^12 +^4 =16.