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

278 Chapter 8 Matrix algebra
8.7 DETERMINANTS
⎛ ⎞
a 11
a 12
a 13
IfA= ⎝a 21
a 22
a 23
⎠, the value of itsdeterminant, |A|, isgiven by
a 31
a 32
a 33
∣ ∣ ∣ ∣ ∣ ∣ ∣∣∣ a
|A| =a 22
a ∣∣∣ ∣∣∣ 23
a
11
−a 21
a ∣∣∣ ∣∣∣ 23
a
a 32
a 12
+a 21
a ∣∣∣ 22
33
a 31
a 13
33
a 31
a 32
If we choose an element ofA, a ij
say, and cross out its row and column and form the
determinant of the four remaining elements, this determinant is known as theminor of
the elementa ij
.
Amoment’s study will therefore reveal thatthe determinant ofAisgiven by
|A| = (a 11
×its minor) − (a 12
×its minor) + (a 13
×its minor)
This method of evaluating a determinant isknown asexpansion along the first row.
Example8.25 Find the determinant of the matrix
⎛ ⎞
121
A = ⎝−134⎠
512
Solution The determinant ofA, written as
121
−134
∣ 512∣
isfound by expanding along itsfirst row:
|A|=1
∣ 3 4
∣ ∣ ∣∣∣ 1 2∣ −2 −1 4
∣∣∣ 5 2∣ +1 −1 3
5 1∣
= 1(2) −2(−22) +1(−16)
= 2+44−16
= 30
Example8.26 Find the minors ofthe elements 1 and 4 inthe matrix
⎛ ⎞
723
B = ⎝103⎠
042
Solution To find the minor of 1 delete its row and column to form the determinant
∣ 2 3
4 2∣ . The
required minor istherefore 4 −12 = −8.
Similarly, the minor of 4 is
∣ 7 3
1 3∣ =21−3=18.