That’s an interesting question. According to the search results, Gaussian elimination has a cubic complexity, which means it takes O(n^3) time to solve a system of n equations with n unknowns, where O is the big O notation. Cramer’s rule, on the other hand, has a factorial complexity, which means it takes O(n!) time to solve the same system, where n! is the factorial of n. However, Cramer’s rule can be realized in far lower complexity than O(n!) by using some clever techniques to calculate the determinants. The best complexity of Cramer’s rule that I could find is O(n^4), which is still worse than Gaussian elimination.

Therefore, from the perspective of time complexity, Gaussian elimination is better than Cramer’s rule. However, there may be other factors that affect the performance of these methods, such as numerical stability, memory usage, and parallelizability. For example, Cramer’s rule may be more suitable for sparse matrices or parallel computing.

As for space complexity, which measures how much memory is required by an algorithm, I could not find any clear answer in the search results. However, I think both methods would have a similar space complexity of O(n^2), since they both need to store an n by n matrix. Gaussian elimination may need some extra space to perform row operations, but that would not change the asymptotic behavior of the space complexity.
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