Page 26 - A Course in Linear Algebra with Applications

10 Chapter One: Matrix Algebra

Thus already we recognise some interesting features of
matrix multiplication. The matrix product is not commuta-
tive, that is, AB and BA may be different when both are de-
fined; also the product of two non-zero matrices can be zero,
a phenomenon which indicates that any theory of division by
matrices will face considerable difficulties.
Next we show how matrix mutiplication provides a way of
representing a set of linear equations by a single matrix equa-
tion. Let A = (aij)m tn and let X and B be the column vectors
with entries x±, X2, ..., x n and 61, b 2, ..., b m respectively. Then
the matrix equation

AX = B

is equivalent to the linear system

• •
{ aiixx + ai 2x 2 + • • • • + + CL 2nXn = = h b x
a\ nx n
CL21X1
CI22X2
+
+
o-mixi + a m2X2 + • • • + a
For if we form the product AX and equate its entries to the
corresponding entries of B, we recover the equations of the
linear system. Here is further evidence that we have got
the definition of the matrix product right.

Example 1.2.4
The matrix form of the pair of linear equations

J 2xi — 3x2 + 5^3 = 1
\ -xi + x 2 - X3 = 4

is