Page 548 - Modelling in Transport Phenomena A Conceptual Approach

528 APPENDIX A. MATHEMATICAL PRELIMINARIES

The inverse of a matrix A is defined by

(A.9-29)

where AdjA is called the adjoint of A. It is obtained from a square matrix A by
replacing each element by its cofactor and then interchanging rows and columns.

Example A.12 Find the inverse of the matrix A given in Example A.11.

Solution
The minor of A is given by

3 1 3 2

-1 1 5
-1 1 1
-2 -2 2
01

1 ;'I
The cofactor matrix is
-1 -1 5
Aij = [ 1
-2 2
The transpose of the cofactor matrix gives the adjoint of A as
-1 1
AdjA= [ ;I 1 i2]
-1 2

Since IAl = 4, the use of Eq. (A.9-29) gives the inverse of A in the form
-0.25
Adj A
A-1 = - 0.25
-
0.25
-0.25
IAl 1.25 -0.25 0.5
A.9.4 Solution of Simultaneous Algebraic Equations
Consider the system of n nonhomogeneous algebraic equations

a1121 + a1222 + ... + al,z, = c1
a2121 + a2222 -t- ... + aznxn = cz
........................................... - (A.9-30)
- ...
anlX1 + an222 f e.. + ann~n = Cn