Matrices: Equivalent Definitions of Singularity I am looking to know as many definitions that are equivalent to the definition of a singular matrix as possible - that is conditions that imply singularity.
Trivially we have the widely known definition: Let $A$ be a $n \times n$ matrix. $A$ is non-singular if $A^{-1}$ exists.
We also have: Let $A$ be a $m \times n$ matrix. $A$ is singular if $m \ne n$
One last example: Let $A$ be a $n \times n$ skew-symmetric matrix; $A$ is singular if $n$ is odd.
With that being said, what other conditions on a matrix $A$ implies singularity that may not be so obvious?
As always, I appreciate any and all contributions made.

Something that may not be as obvious is that if $A$ is $n\times n,$ and its rows or columns are linearly dependent.
This is equivalent to saying $A$ has $0$ as an eigenvalue, and additionally that one can apply Gaussian elimination on $A$ and find that it is row (column) equivalent to a matrix with a row (column) of all $0$s.
