Am I misreading this, or is it really about a 2x2 matrix? (In which case computing the determinant involves just two multiplications and one subtraction.)
The problem with the determinant is not about performance. It is just useless for determining if a matrix is singular. The thing that gives it away is that the determinant is influenced by a rescaling of the matrix:
det(s A) = s^n det(A) where A is a n x n matrix
As an example, would you say that [[1e-10, 0], [0, 1e-10]] is singular? It has condition number 1.
Let's say we are in a setting where we only work with integers. A matrix is invertible iff its determinant is invertible in the underlying ring. The only invertible elements in Z are -1 and 1.
So, the code is also incorrect in the integer setting. Here, we should not check for 0, but for -1 or 1.
This is purely a matter of numerical stability. Of course [[1e-10, 0], [0, 1e-10]] is nonsingular, and its determinant is 1e-20, which does not equal zero.
Yes, when it comes to floating point issues we might want to use something else, and that's a valid complaint when it comes to NumPy code, but from a theoretical perspective the determinant is an excellent tool to determine singularity.
Numerically, such a binary answer is pretty useless. Here, we need a measure of how singular/nonsingular a matrix is relative to the numerical precision we are working with.