Engineering Fundamentals: An Introduction to Engineering, 4th ed.c

612 Chapter 18 Mathematics in Engineering

Expressing Equations (18.14a) and (18.14b) in a matrix form, we have

[A]

To solve for the unknownsx 1 andx 2 , we may first solve forx 2 in terms ofx 1 , using Equa-

tion (18.14 b), and then substitute that relationship into Equation (18.14a). These steps are

shown next.

Solving forx 1 :

(18.15a)

After we substitute forx 1 in either Equation (18.14a) or (18.14b), we get

(18.15b)

Referring to the solutions given by Equations (18.15a) and (18.15b), we see that the denomi-

nators in these equations represent the product of coefficients in the main diagonal minus the

product of the coefficient in the other diagonal of the [A] matrix. Thea 11 a 22 – a 12 a 21 is the

determinantof the 2 2 [A] matrix and is represented in one of following ways:

(18.16)

Only the determinant of a square matrix is defined. Moreover, keep in mind that the determi-

nant of the [A] matrix is a single number. That is, after we substitute for the values ofa 11 ,a 22 ,

a 12 , anda 21 intoa 11 a 22 – a 12 a 21 , we get a single number.

Let us now consider the determinant of a 3 by 3 matrix such as

which is computed in the following manner:

(18.17)

There is a simple procedure calleddirect expansionthat you can use to obtain the results given

by Equation (18.17). Direct expansion proceeds in the following manner. First we repeat and

place the first and the second columns of the matrix [C] next to the third column, as shown in

†

c 11 c 12 c 13

c 21 c 22 c 23

c 31 c 32 c 33

†c 11 c 22 c 33 c 12 c 23 c 31 c 13 c 21 c 32 c 13 c 22 c 31 c 11 c 23 c 32 c 12 c 21 c 33

3 C 4 £

c 11 c 12 c 13

c 21 c 22 c 23

c 31 c 32 c 33

§

Det 3 A 4 or det 3 A 4 or `

a 11 a 12

a 21 a 22

`a 11 a 22 a 12 a 21

x 2 

a 11 b 2 b 1 a 21

a 11 a 22 a 12 a 21

x 1 

b 1 a 22 a 12 b 2

a 11 a 22 a 12 a 21

x 2 

b 2 a 21 x 1

a 22

1 a 11 x 1 a 12 a

b 2 a 21 x 1

a 22

bb 1

c

a 11 a 12

a 21 a 22

de

x 1

x 2

fe

b 1

b 2

f

u

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