I want to find the best algorithm to determine if an $n \times n$ matrix is invertible in high dimensions...
Is the best way to determine the invertibility of a matrix is to calculate the determinant of that matrix? If the determinant is $0$, then it is not invertible, and if it is non-zero, then it is invertible... right? I also found a theorem that I did not understand how to use it:
An $n \times n$ matrix $A$ is invertible if and only if it is row equivalent to the $n \times n$ identity matrix $I$.