Page 136 - Numerical Analysis Using MATLAB and Excel

Chapter 4 Matrices and Determinants

Property 3:

If two rows or two columns of a matrix are identical, the determinant is zero. This follows from Prop-
erty 2 with m = . 1

4.6 Cramer’s Rule
Let us consider the systems of the three equations below

a x + a y + a z = A
11
13
12
a x + a y + a z = B (4.30)
22
21
23
a x + a y + a z = C
31
33
32
and let
a 11 a 12 a 13 Aa 11 a 13 a 11 Aa 13 a 11 a 12 A
Δ = a 21 a 22 a 23 D = Ba 21 a 23 D = a 21 Ba 23 D = a 21 a 22 B
2
1
3
a 31 a 32 a 33 Ca 31 a 33 a 31 Ca 33 a 31 a 32 C
Cramer’s rule states that the unknowns , , and can be found from the relations
xy
z
D D D
x = ------ 1 y = ------ 2 z = ------ 3 (4.31)
Δ Δ Δ
provided that the determinant Δ (delta) is not zero.

We observe that the numerators of (4.31) are determinants that are formed from Δ by the substi-
tution of the known values A, B, and C, for the coefficients of the desired unknown.
Cramer’s rule applies to systems of two or more equations.

If (4.30) is a homogeneous set of equations, that is, if A = B = C = 0 , then, D D and D, 1 2 , 3 are
all zero as we found in Property 1 above. Then, x = y = z = 0 also.

Example 4.10

Use Cramer’s rule to find v v and v, 1 2 , 3 if

2v – 5 – v + 3v = 0
2
1
3
– 2v – 3v – 4v = 8 (4.32)
3
1
2
v + 3v – 4 – v = 0
3
1
2
and verify your answers with MATLAB.
4−18 Numerical Analysis Using MATLAB® and Excel®, Third Edition

Copyright © Orchard Publications