Given the system of equations $AX=B$ where $A$ is singular. Can one solve for $X$ in this case? Consider the following example for illustratrion
$A = \begin{pmatrix} 0 & a_3 & -a_2\\
-a_3 & 0 & a_1 \\
a_2 & -a_1 & 0 \end{pmatrix}$,
$X= \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ and $B = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} $.
Edit: There was a typo in $A$.
How can one solve for $x,y,z$ in this case?
Best Answer
If a matrix is singular it means that its determinant is zero.
If a determinant is zero it means some row/col is a linear combination of other rows/cols.
So, not all vectors ${x,y,z}$ can be expressed as a combination of the vectors that each row/col of the matrix represents (The matrix is a tranformation between bases).
In general you can not solve the system.
But there are cases where the matrix is able to represent some group of vectors. Using a submatrix obtained by eliminating the linear combination, and that submatrix not being singular.
Suppose you eliminated the third column. Then, you can still obtain $x,y$, but $z$ coordinate will be undefined.