In this question Necessity/Advantage of LU Decomposition over Gaussian Elimination it is asked why LU factorization is useful. I understand how this reduces time complexity of solving a number equations of the form Ax=b for matrix A and column matrix b but why don't you just find A-1 instead?
Inversion has a lower time complexity than LU factorization (comparing the value used in the previous link and ones found here https://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations) and matrix multiplication has the same time complexity in this case as is needed to solve for different values of b.
Overall, I see the value of LU factorization as opposed to resolving multiple matrix equations but I don't know why it would be better than the method I described that uses matrix inversion. Clearly LU factorization has some value, I would like to know what that it. Thanks
I believe the answer to this question is that all square matrices have a PTLU factorization while not all square matrices are invertible. Therefore PTLU factorization is more versatile. Any other insights are still appreciated however so please comment or answer the question. Thanks