0195136047.pdf

SOLUTION OF SIMULTANEOUS EQUATIONS 845

c 22 x 2 +c 23

d 3

c 33

=d 2

or x 2 =

1

c 22

[

−c 23

d 3

c 33

+d 2

]

(8b)

Substituting Equations (8a) and (8b) into Equation (7a), one can solve forx 1.

MATRIX METHOD

For those readers who have been introduced to matrix methods of analysis, the set of equations

in Equation (4) can be expressed as

AX=B (9)

where

coefficient matrix,A=





a 11 a 12 a 13

a 21 a 22 a 23

a 31 a 32 a 33



 (10a)

column matrix of unknowns,X =





x 1

x 2

x 3



 (10b)

and

column matrix,B=





b 1

b 2

b 3



 (10c)

The solution is accomplished by

X=A−^1 B (11)

whereA−^1 is the inverse of the matrixA.

One can find the inverse of a matrix by following the steps

1. Obtain thedeterminantof the matrixAas indicated in Equation (6a); it should be nonzero
   for the inverse to exist.


2. Replace each element of the matrix by itscofactor. For example, the cofactor ofa 11 in
   Equation (10a) is given by

(− 1 )^1 +^1

∣

∣∣

∣

a 22 a 23

a 32 a 33

∣

∣∣

∣=(a^22 a^33 −a^23 a^32 );

the cofactor ofa 12 in Equation (10a) is given by

(− 1 )^1 +^2

∣

∣

∣

∣

a 21 a 23

a 31 a 33

∣

∣

∣

∣=−(a^21 a^33 −a^23 a^31 )

3. Find thetransposeof the resultant matrix in STEP 2 by interchanging its rows and columns.


4. Divide the resultant matrix in STEP 3 by the determinant found in STEP 1.

As a check, one can verify that the productAA−^1 results in aunit matrix.