Understanding Engineering Mathematics

(やまだぃちぅ) #1
=adeh+bcfg−bceh−adfg
=(ad−bc)(eh−fg) ( 41

)

=





ab
cd









ef
gh





as required.


Exercises on 13.4



  1. Evaluate (i)






32
− 11




∣ (ii)





33
11






  1. Expand by (i) first row (ii) second column (iii) second row







320
107
241







  1. Simplify and evaluate







910 10
23 − 1
331






Answers



  1. (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 3

If 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

∣ ∣ ∣ ∣ ∣ 
Free download pdf