s
Section 5.7 Matrices
447
The constraints in this problem are represented by the system of linear
equations
x + y = 70
24x + 14y = 1180
These are linear equations because each of the two unknowns, x and y, appears
only to the first power. (Also an equation such as ax + by = c represents a straight
line when graphed on an x–y coordinate system.) The solution to this system of
equations is x = 20, y = 50 (you can easily check that this is a solution). The
matrix
a = c
1
1
24 14
d
is the matrix of coefficients for this system of linear equations. As we will see, the
matrix of coefficients can be used to solve a system of linear equations.
■
PRaCtiCe 50 In the matrix:
a = c
1 4 −6
8
3 0
1 −7
d
What is a 23 ? What is a 24 ? What is a 13 ?
In a matrix, the arrangement of the entries is significant. Therefore, for two
matrices to be equal, they must have the same dimensions and the same entries
in each location.
example 64
Let
X = £
x 4
1 y
z 0
§
Y = £
3 4
1 6
2 w
§
If X = Y, then x = 3, y = 6, z = 2, and w = 0.
We will often be interested in square matrices, in which the number of rows
equals the number of columns. If a is an n × n square matrix, then the elements
a 11 , a 22 , … , a nn form the main diagonal of the matrix. If the corresponding elements match when we think of folding the matrix along the main diagonal, then
the matrix is symmetric about the main diagonal. In a symmetric matrix, a ij = a ji .