# 4.23: Untitled Page 78

- Page ID
- 18211

## Chapter 4

(4‐117)

while the general element *c * is given by

*ij*

(4‐118)

In Eq. 4‐116 we see that an *m * *n * matrix can multiply an *n * *p * to produce an *m * *p *, and we see that the matrix multiplication represented by AB is *only* *defined* when the number of columns in A is equal to the number of rows in B .

The matrix multiplication illustrated in Eq. 4‐114 conforms to this rule since there are *three columns* in the matrix of mass fractions and *three rows* in the column matrix of mass flow rates. The configuration illustrated in Eq. 4‐114 is extremely common since it is associated with the solution of *n* equations for *n* unknowns.

Our generic representation for this type of matrix equation is given by Au b

(4‐119)

in which A is a *square matrix*, *u* is a column matrix of *unknowns*, and *b* is a column matrix of *knowns*. These matrices are represented explicitly by

11

*a*

12

*a*

*...... *

1

*a n *

1

*u *

1

*b *

*a*

*a*

*...... a*

*. *

*. *

21

22

2 *n*

A

*, *

u

*, *

b

(4‐120)

*.... *

*.... *

*...... *

*.... *

*. *

*. *

*a * 1

*n*

*n*

*a * 2 *...... *

*n*

*a n*

*n*

*u *

*n*

*b *

Sometimes the coefficients in A depend on the unknowns, u, and the matrix equation may be *bi‐linear* as indicated in Eqs. 4‐91.

The *transpose* of a matrix is defined in the same manner as the transpose of an array that was discussed in Sec. 2.5, thus the transpose of the matrix A is constructed by interchanging the rows and columns to obtain

11

*a*

*a* 21 *... a * 1

*m *

11

*a*

12

*a*

*...... *

1

*a n *

12

*a*

*a* 22 *... *

*m*

*a * 2

*a*

*a*

*...... a*

21

22

2 *n*

T

A

*, *

A

*. *

*. *

*... *

*. *

(4‐121)

*. *

*. *

*...... *

*. *

*. *

*. *

*... *

*. *

*a*

1

*m*

*m*

*a * 2 *...... *

*m*

*a n*

1

*a n a* 2 *n ... *

*m*

*a n*

Here it is important to note that A is an *m * *n * matrix while T

A is an *n * *m *

matrix.

138

*4.9.1 Inverse of a square matrix *

In order to solve Eq. 4‐119, one cannot “divide” by A to determine the unknown, u, since *matrix division is not defined*. There is, however, a related operation involving the *inverse* of a matrix. The inverse of a matrix, A , is another

matrix,

1

A

, such that the product of A and

1

A

is given by

1

A A

I

(4‐122)

in which I is the *identity matrix*. Identity matrices have ones in the diagonal elements and zeros in the off‐diagonal elements as illustrated by the following 4 4 matrix:

1 0 0 0

0 1 0 0

I

(4‐123)

0 0 1 0

0 0 0 1

For the inverse of a matrix to exist, the matrix must be a *square matrix*, i.e., the number of rows must be equal to the number of columns. In addition, the *determinant* of the matrix must be different from zero. Thus for Eq. 4‐122 to be valid we require that the determinant of A be different from zero, i.e., A

0

(4‐124)

This type of requirement plays an important role in the derivation of the *pivot* *theorem* (see Sec. 6.4) that forms the basis for one of the key developments in Chapter 6.

As an example of the use of the inverse of a matrix, we consider the following set of four equations containing four unknowns:

(4‐125)

In compact notation, we would represent these equations according to Au

b

(4‐126)

139