082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

8.7 Determinants 279
In addition to finding the minor of each element in a matrix, it is often useful to find a
related quantity -- the cofactor of each element. The cofactor is found by imposing on
the minor a positive or negative sign depending upon its position, that is a place sign,
according tothe following rule:
+ − +
− + −
+ − +
Example8.27 If
⎛ ⎞
3 27
A = ⎝9 10⎠
3−12
find the cofactors of 9 and 7.
Solution The minor of 9 is
2 7
∣−1 2∣ = 4 − (−7) = 11, but since its place sign is negative, the
required cofactor is −11.
The minor of 7 is
∣ 9 1
3 −1∣ = −9 − 3 = −12. Its place sign is positive, so that the
required cofactor issimply −12.
8.7.1 Usingdeterminantstofindvectorproducts
Determinants can also be used to evaluate the vector product of two vectors. If a =
a 1
i +a 2
j +a 3
k and b = b 1
i +b 2
j +b 3
k, we showed in Section 7.6 that a × b is the
vector defined by
a×b = (a 2
b 3
−a 3
b 2
)i+ (a 3
b 1
−a 1
b 3
)j+ (a 1
b 2
−a 2
b 1
)k
If weconsider the expansion of the determinant given by
∣ i jk ∣∣∣∣∣
a 1
a 2
a 3
∣b 1
b 2
b 3
we find the same result. This definition is therefore a convenient mechanism for evaluating
a vector product.
Ifa =a 1
i+a 2
j+a 3
kandb =b 1
i+b 2
j+b 3
k,then
∣ i jk ∣∣∣∣∣
a×b=
a 1
a 2
a 3
∣b 1
b 2
b 3