How can I recover a variable with singular coefficent

Question

How can I recover $X$ from

\begin{equation} AX = B \end{equation}

when $A$ is a singular matrices?

Answer 1

In general you can't. For example, both vectors

$$\,X_1=\binom{1}{0}\;\;,\;\;X_2=\binom{1}{1}$$

are solutions to the system

$$A_1=\begin{pmatrix}1&0\\0&0\end{pmatrix}\binom{x}{y}=\binom{1}{0}$$

Answer 2

For most matrix equations, actually there is a unique solution. You just assign variables to the contents of $X$ and solve. You will get a system of linear equations. The above example had multiple solutions because one of the equations was $0x+0y=0$, which is always true.