Currently using the MIT recordings from Prof.Strang to review deduction of the determinant formula.
Let's say we take a closer look at
$$
\left\vert\begin{array}{c c}
a & 0\\
c & 0
\end{array}\right\vert
$$
And to quote from the transcript:
Why is that determinant nothing, forget him?
Well, it has a column of zeros.
And by the — well, so one way to think is, well, it's a singular matrix.
Oh, for, for like forty-eight different reasons. That determinant is zero.
Now $48$, that's an oddly specific number. And while I do believe he was exaggerating, it did make me curious about just how many reasons (the determinant's being zero not included, of course) there really are for that matrix' being singular.
What are they?
Best Answer
Here are $23$ equivalent conditions for a matrix to be nonsingular; just take the negation of each statement and you have $23$ conditions for a matrix to be singular:
http://mathworld.wolfram.com/InvertibleMatrixTheorem.html