Linear Algebra, 2020a

Section I. Solving Linear Systems 31
3.12 Example The first of these matrices is nonsingular while the second is
singular
( ) ( )
1 2 1 2
3 4 3 6
because the first of these homogeneous systems has a unique solution while the
second has infinitely many solutions.
x + 2y = 0
3x + 4y = 0
x + 2y = 0
3x + 6y = 0
We have made the distinction in the definition because a system with the same
number of equations as variables behaves in one of two ways, depending on
whether its matrix of coefficients is nonsingular or singular. Where the matrix
of coefficients is nonsingular the system has a unique solution for any constants
on the right side: for instance, Gauss’s Method shows that this system
x + 2y = a
3x + 4y = b
has the unique solution x = b−2a and y =(3a−b)/2. On the other hand, where
the matrix of coefficients is singular the system never has a unique solution — it
has either no solutions or else has infinitely many, as with these.
x + 2y = 1
3x + 6y = 2
x + 2y = 1
3x + 6y = 3
The definition uses the word ‘singular’ because it means “departing from
general expectation.” People often, naively, expect that systems with the same
number of variables as equations will have a unique solution. Thus, we can think
of the word as connoting “troublesome,” or at least “not ideal.” (That ‘singular’
applies to those systems that never have exactly one solution is ironic but it is
the standard term.)
3.13 Example The systems from Example 3.3, Example 3.4, and Example 3.8
each have an associated homogeneous system with a unique solution. Thus these
matrices are nonsingular.
( )
3 4
2 −1
⎛
⎜
3 2 1
⎞
⎟
⎝6 −4 0⎠
0 1 1
⎛
⎜
1 2 −1
⎞
⎟
⎝2 4 0 ⎠
0 1 −3
The Chemistry problem from Example 3.5 is a homogeneous system with more