Page 543 - Modelling in Transport Phenomena A Conceptual Approach

A.9. MATRTCES 523

A.9.1 Fundamental Algebraic Operations

1. Two matrices A = (aij) and B = (bij) of the same order are equal if and
only if aij = bij.

2. If A = (aij) and B = (bij) have the same order, the sum of A and B is
defined as
A + B = (aij + bij) (A.9-3)
If A, B, and C are the matrices of the same order, addition is commutative
and associative, Le.,
A+B=B+A (A.9-4)

A + (B + C) = (A + B) + C (A.9-5)

3. If A = (aij) and B = (bij) have the same order, the difference of A and B is
defined as
A - B = (aij - bij) (A.9-6)

Example A.9 If
A=[; 2 -1 Cl] andB=[i 2 -4 ;]

determine A + B and A - B.

Solution

0+0 I=[! T]
2+2 -1-4
A+B = [ 1+3
5+1
3+0 0-0 ]= [ a]
2-2 -1+4
A-B = [ 1-3 :2
3-0 5-1

4. If A = (aij) and X is any number, the product of A by X is defined as

XA = AX = (Xaij) (A.9-7)

5. The product of two matrices A and B, AB, is defined only if the number
of columns in A is equal to the number of rows in B. In this case, the two