So
x 2 =a 11 b 2 −a 21 b 1
a 11 a 22 −a 21 a 12Similarly we find
x 1 =b 1 a 22 −a 12 b 2
a 11 a 22 −a 21 a 12Introduce the notation (337
➤ ) ∣ ∣ ∣ ∣ab
cd∣
∣
∣
∣=ad−bcthen 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∣∣
∣
∣Sox 2 =3 × 2 − 1 × 1
3 ×(− 2 )− 1 × 2=−5
8Similarly we findx 1 =1 (− 2 )− 2 × 2
3 (− 2 )− 1 × 2=3
4Introduce the notation
∣
∣
∣
∣32
1 − 2∣
∣
∣
∣=^3 (−^2 )−^1 ×^2then we can writex 1 =∣
∣
∣
∣12
2 − 2∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣32
1 − 2∣
∣
∣
∣=3
4x 2 =∣
∣
∣
∣31
12∣ ∣ ∣ ∣ ∣ ∣∣∣^32
1 − 2∣
∣
∣∣=−5
8Notice 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 thoughtof 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 ,...