The nonhomogenous system Ax=b

[Bipolarity]

The invertible matrix equation tells us that the following statements are equivalent, for any square matrix A:
1) A is invertible
2) Ax=0 has only the trivial solution
3) Ax=b has a unique solution for any column vector b

My question:
Suppose you know that A is a singular matrix. Then can you conclude that for every column vector b, either Ax=b has no solution or Ax=b has infinitely many solutions? Or is there a vector b for which Ax=b has a unique solution, despite the fact that A is singular?

Also, given that A is singular, how can you tell whether Ax=b has no solutions or has infinitely many solutions?

BiP

 

[CompuChip]

If A is not invertible then it is not true that (Ax = b has a unique solution for any column vector b).
Negating that statement,
$\neg\left( \forall b \in \text{column vectors}, Ax = b \text{ has a unique solution } \right)$
gives
$\exists b \in \text{column vectors}, \neg \left( Ax = b \text{ has a unique solution } \right)$
i.e.
for at least one column vector b, Ax = b does not have a unique solution.

It doesn't say anything about any column vector b, and it also doesn't say whether "not a unique solution" means "no solution at all" or "infinitely many".

One way to check is to form the augmented matrix
$$[A | b] = \left[\begin{array}{ccc|c}
A_{11} & \cdots & A_{1n} & b_1 \\
\vdots & \ddots & \vdots & b_2 \\
A_{m1} & \cdots & A_{mn} & b_3
\end{array}\right].$$

If you row-reduce it, you will end up with one or more rows of zeroes at the bottom. If the corresponding entries in the right-most column are non-zero, there are no solutions; otherwise there are infinitely many.

 

[HallsofIvy]

If A is not invertible, then its "kernel", the set of all x such that Ax= 0, is non-trivial. It is a subspace of the domain of A with dimension at least 1. Let $x_0$ be a non-zero vector such that $Ax_0= 0$. If Ax= b has a solution, that is, if there exist $x_1$ such that $Ax_1= b$ then $A(x_1+ x_0)= Ax_1+ Ax_0= b+ 0= b$ so $x= x_1+ x_0$ is another vector such that Ax= b.

Do you know what a "linear manifold" is? In R², a subspace is a line through the origin. A linear manifold is a line NOT through the origin. In R³ a two dimensional subpace is a plane through the origin. A two dimensional linear manifold is a plane NOT containing the origin. Given a linear manifold, we can take any vector v in it and show that the set {x- v| x in the linear manifold} is a subspace. We say that the linear manifold is "parallel"
to the kernel.

If A is not invertible, then its kernel, {x| Ax= 0}, is a subspace of dimension at least 1. And if Ax= b has at least one solution then {x|Ax= b} is a linear manifold of the same dimension as the kernel and "parallel" to it.

In particular, if A is not invertible and Ax= b has a solution then there are an infinite number of solutions.