Page 105 - Matrix Analysis & Applied Linear Algebra

3.5 Matrix Multiplication 99

Matrix multiplication provides a convenient representation for a linear sys-
tem of equations. For example, the 3 × 4 system
2x 1 +3x 2 +4x 3 +8x 4 =7,
3x 1 +5x 2 +6x 3 +2x 4 =6,
4x 1 +2x 2 +4x 3 +9x 4 =4,

can be written as Ax = b, where
 
  x 1  
2348 7
6
 x 2 
A 3×4 =  3562  , x 4×1 =   , and b 3×1 =   .
x 3
4249 4
x 4
And this example generalizes to become the following statement.
Linear Systems

Every linear system of m equations in n unknowns

a 11 x 1 + a 12 x 2 + ··· + a 1n x n = b 1 ,
a 21 x 1 + a 22 x 2 + ··· + a 2n x n = b 2 ,
.
. .
a m1 x 1 + a m2 x 2 + ··· + a mn x n = b m ,

can be written as a single matrix equation Ax = b in which

     
a 11 a 12 ··· a 1n x 1 b 1
 a 21 a 22 ··· a 2n   x 2   b 2 
A =  . . . .  , x =  .  , and b =  .  .
 .
.
.
. . . . . . .   .   . 
a m1 a m2 ··· a mn x n b m
Conversely, every matrix equation of the form A m×n x n×1 = b m×1 rep-
resents a system of m linear equations in n unknowns.

The numerical solution of a linear system was presented earlier in the text
without the aid of matrix multiplication because the operation of matrix mul-
tiplication is not an integral part of the arithmetical process used to extract a
solution by means of Gaussian elimination. Viewing a linear system as a single
matrix equation Ax = b is more of a notational convenience that can be used to
uncover theoretical properties and to prove general theorems concerning linear
systems.