Page 82 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 3 MATRICES 59

Multiplication or Division. Multiplication or division by a constant:

Multiplication of two matrices can be either scalar or matrix multiplication.
Scalar multiplication of two matrices consists of multiplying the elements of a
matrix by a constant, as shown above, or multiplying corresponding elements of
two matrices:
[l h [: !]=[:::
b c
axr bxs cxt
AxB= d e f x u v iii

Thus it's clear that both matrices must have the same dimensions m x n.
Scalar multiplication is commutative, that is, A x B = B x A.

Matrix Multiplication. The matrix multiplication of two matrices is
somewhat more complicated. The individual matrix elements of the matrix
product C of two matrices A and B are
n
c, = zAikBb
k=l
where i is the row number andj is the column number. Thus, for example,
ar+bu+cx as+bv+cy at+bw+cz
dr+eu+fjc ds+ev+fL dt+ew+fi
gr + hu + ix gs + hv + iy gt + hw + iz

Matrix multiplication is not generally commutative, that is A.B # B.A.

Transposition. The transpose of matrix A, most commonly written as AT, is
the matrix obtained by exchanging the rows and columns of A; that is, the matrix
element aij becomes the element aji in the transposed matrix. The transpose of a
matrix of N rows and Mcolumns is a matrix of M rows and N columns.
Matrix Inversion. The process of matrix inversion is analogous to obtaining
the reciprocal of a number a. The matrix relationship that corresponds to the
algebraic relationship a x (l/a) = 1 is

AA-'=I