Page 23 - A Course in Linear Algebra with Applications

1.2: Operations with Matrices 7

Example 1.2.1
If
, / 1 2 0\ , „ (I 1 1
A dB
={-1 0 l) ™ ={ 0 -3 1
then

3
2A + 5 = [ X A " I and 2A - 35

(iii) Matrix multiplication
It is less obvious what the "natural" definition of the
product of two matrices should be. Let us start with the
simplest interesting case, and consider a pair of 2 x 2 matrices

an a 12\ , , D ( b lx b X2
lL
A= ( ^ and B - , , .
\ G 2 1 0122/ \ 0 2 1 022
In order to motivate the definition of the matrix product AB
we consider two sets of linear equations

a\iVi + a.i2V2 = xi a n d f &nzx + b X2z 2 = y±
o.2iV\ + a 22y 2 = x 2 \ b 21zi + b 22z 2 = y 2

Observe that the coefficient matrices of these linear systems
are A and B respectively. We shall think of these equations
as representing changes of variables from j/i, y 2 to xi, x 2, and
from z\, z 2 to y\, y 2 respectively.
Suppose that we replace y\ and y 2 in the first set of equa-
tions by the values specified in the second set. After simplifi-
cation we obtain a new set of equations

(aii&n + ai 2b 2i)zi + (a U 0 1 2 + a i2b 22)z 2 = %i
(a 21bn + a 22b 2i)zi + (a 2ib 12 + a 22b 22)z 2 = x 2