=adeh+bcfg−bceh−adfg
=(ad−bc)(eh−fg) ( 41
➤
)=∣
∣
∣
∣ab
cd∣
∣
∣
∣∣
∣
∣
∣ef
gh∣
∣
∣
∣as required.
Exercises on 13.4
- Evaluate (i)
∣
∣
∣
∣32
− 11∣
∣
∣
∣ (ii)∣
∣
∣
∣33
11∣
∣
∣
∣- Expand by (i) first row (ii) second column (iii) second row
∣
∣
∣
∣
∣320
107
241∣
∣
∣
∣
∣- Simplify and evaluate
∣
∣
∣
∣
∣910 10
23 − 1
331∣
∣
∣
∣
∣Answers
- (i) 5 (ii) 0 2.−58 in each case 3.− 26
13.5 Cramer’s rule for solving a system of linear equations
We will now show how to generalise the 2×2 version of Cramer’s rule given in
Section 13.4 by looking at the case of three equations in three unknowns.
a 11 x 1 +a 12 x 2 +a 13 x 3 =b 1
a 23 x 1 +a 22 x 2 +a 23 x 3 =b 2
a 31 x 1 +a 32 x 2 +a 33 x 3 =b 3If we solve this system by elimination then it is found (believe me!) that the solution can
be expressed in a simple determinant form that generalises the 2×2 case of Section 13.4.
This isCramer’s rule. Cramer’s rule gives the solution in terms of 3×3 determinants as:
x 1 =∣
∣
∣
∣
∣b 1 a 12 a 13
b 2 a 22 a 23
b 3 a 32 a 33∣ ∣ ∣ ∣ ∣ x 2 =∣
∣
∣
∣
∣a 11 b 1 a 13
a 21 b 2 a 23
a 31 b 3 a 33∣ ∣ ∣ ∣ ∣ x 3 =∣
∣
∣
∣
∣a 11 a 12 b 1
a 21 a 22 b 2
a 31 a 32 b 3∣ ∣ ∣ ∣ ∣