Singular vs. Non-singular I'm asking this for clarification. A matrix/system of equations is singular is there are infinite solutions, but iff there is a unique solution then its non-singular?
I haven't learned how to take a determinant yet. However, my professor went over how to determine if it is singular or non-singular without needing to take it. I've looked in my notes, but haven't been able to find it.
Any clarity would be much appreciated. 
 A: Suppose the linear system we have is 
\begin{equation}
A x = b
\end{equation}
where $\newcommand{\reals}{{\mathbf R}}A\in\reals^{n\times n}$ and $x, b \in\reals^n$.
You need to be a bit more precise to be correct to relate the number (or existence) of solutions to the singularity of $A$.
The following statements are correct:


*

*A linear system has a unique solution if and only if the matrix is non-singular.

*A linear system has either no solution or infinite number of solutions if and only if the matrix is singular.

*A linear system has a solution if and only if $b$ is in the range of $A$.


Now by definition,


*

*The matrix is non-singular if and only if the determinant is nonzero.


However, like your professor mentioned, you do not need to evaluate the determinant to see whether a matrix is singular or not (though most such methods evaluates the determinant as by-product).
For example, you can use Gaussian elimination to tell whether a matrix is singular. This has the following advantages.


*

*The time complexity of Gaussian elimination is $O(n^3)$ (whereas brute-force evaluation of determinant by the original definition takes $O(n!)$).

*Gaussian elimination evaluates the determinant as by-product (i.e., with no additional cost).


Hope this helps you!
