Page 15 - Matrix Analysis & Applied Linear Algebra

1.2 Gaussian Elimination and Matrices 7

Finally, substitute z = 1 and y = 2 back into the first equation in (1.2.5) to
get
1 1
x = (1 − y − z)= (1 − 2 − 1) = −1,
2 2
which completes the solution.
It should be clear that there is no reason to write down the symbols such
as “ x, ”“ y, ”“ z, ” and “ = ” at each step since we are only manipulating the
coefficients. If such symbols are discarded, then a system of linear equations
reduces to a rectangular array of numbers in which each horizontal line represents
one equation. For example, the system in (1.2.4) reduces to the following array:

 
211 1
 621 −1  . (The line emphasizes where = appeared.)
−221 7

The array of coefficients—the numbers on the left-hand side of the vertical
line—is called the coefficient matrix for the system. The entire array—the
coefficient matrix augmented by the numbers from the right-hand side of the
system—is called the augmented matrix associated with the system. If the
coefficient matrix is denoted by A and the right-hand side is denoted by b ,
then the augmented matrix associated with the system is denoted by [A|b].
Formally, a scalar is either a real number or a complex number, and a
matrix is a rectangular array of scalars. It is common practice to use uppercase
boldface letters to denote matrices and to use the corresponding lowercase letters
with two subscripts to denote individual entries in a matrix. For example,

 
a 11 a 12 ··· a 1n
 a 21 a 22 ··· a 2n 
A =  . . . .  .
 . . . . 
. . . .
a m1 a m2 ··· a mn
The first subscript on an individual entry in a matrix designates the row (the
horizontal line), and the second subscript denotes the column (the vertical line)
that the entry occupies. For example, if

 
2 1 3 4
A =  8 6 5 −9  , then a 11 =2,a 12 =1,. . . ,a 34 =7. (1.2.6)
−3 8 3 7

A submatrix of a given matrix A is an array obtained by deleting any

2 4
combination of rows and columns from A. For example, B = is a
−3 7
submatrix of the matrix A in (1.2.6) because B is the result of deleting the
second row and the second and third columns of A.