Understanding Engineering Mathematics

(やまだぃちぅ) #1

So


x 2 =

a 11 b 2 −a 21 b 1
a 11 a 22 −a 21 a 12

Similarly we find


x 1 =

b 1 a 22 −a 12 b 2
a 11 a 22 −a 21 a 12

Introduce the notation (337


➤ ) ∣ ∣ ∣ ∣

ab
cd




∣=ad−bc

then we can write


x 1 =





b 1 a 12
b 2 a 22






∣∣

a 11 a 12
a 21 a 22


∣∣

x 2 =





a 11 b 1
a 21 b 2





∣∣


a 11 a 12
a 21 a 22

∣∣


So

x 2 =

3 × 2 − 1 × 1
3 ×(− 2 )− 1 × 2

=−

5
8

Similarly we find

x 1 =

1 (− 2 )− 2 × 2
3 (− 2 )− 1 × 2

=

3
4

Introduce the notation




32
1 − 2




∣=^3 (−^2 )−^1 ×^2

then we can write

x 1 =





12
2 − 2

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

32
1 − 2





=

3
4

x 2 =





31
12

∣ ∣ ∣ ∣ ∣ ∣

∣∣^32
1 − 2



∣∣

=−

5
8

Notice the patterns in the last steps:


(i) The denominator is the same forx 1 andx 2 and clearly obtained from the coefficients
of the left-hand side of the equations.

(ii) The numerator forx 1 is given by replacing thex 1 column of the array of coefficients
by the right-hand side column ofb’s. Similarly, the numerator forx 2 is obtained by
replacing thex 2 column by the right-hand side column.


This rule, capitalising on the notation introduced, is calledCramer’s rule– we will
say more on this in Section 13.5.


The quantity





ab
cd




∣is called a^2 ×^2 orsecond order determinant. It can be thought

of as associated with the matrix


[
ab
cd

]

. We denote the determinant of a matrixAby|A|


or det(A). By solving systems of 3, 4 ,...equations, we are led to definitions of 3, 4 ,...

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