Page 150 - Numerical Analysis Using MATLAB and Excel

Chapter 4 Matrices and Determinants

4.12 Summary

• A matrix is a rectangular array of numbers whose general form is

a 11 a 12 a 13 … a 1n
a 21 a 22 a 23 … a 2n
A = a 31 a 32 a 33 … a 3n

… … ………
a m1 a m2 a m3 … a mn

The numbers a ij are the elements of the matrix where the index indicates the row, and j
i
indicates the column in which each element is positioned. A matrix of rows and columns
m
n
is said to be of m × n order matrix. If m = n , the matrix is said to be a square matrix of order
m .

• Two matrices A = a ij and B = b ij are equal, that is, A = B , if and only if

,
,
,,
,,,
,
a = b ij i = 1 2 3 … m j = 123 … n
ij
•Two matrices are said to be conformable for addition (subtraction), if they are of the same order
m × n . If A = a ij and B = b ij are conformable for addition (subtraction), their sum (dif-
C
A
B
ference) will be another matrix with the same order as and , where each element of C
is the sum (difference) of the corresponding elements of A and B , i.e.,
C = A ± B = [ a ± b ] ij
ij
• If is any scalar (a positive or negative number), and not [ ] which is a 1 × 1 matrix, then
k
k
k
A
A
multiplication of a matrix by the scalar , is the multiplication of every element of by . k
•Two matrices and are said to be conformable for multiplication A B⋅ in that order, only
B
A
when the number of columns of matrix is equal to the number of rows of matrix . That is,
B
A
the product AB⋅ (but not BA⋅ ) is conformable for multiplication only if is an m × p and
A
matrix is an p × n matrix. The product AB⋅ will then be an m × n matrix.
B
• For matrix multiplication, the operation is row by column. Thus, to obtain the product AB⋅ ,
we multiply each element of a row of by the corresponding element of a column of ; then,
B
A
we add these products.
• Division of one matrix by another, is not defined.

4−32 Numerical Analysis Using MATLAB® and Excel®, Third Edition

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