human
Which is better for solving systems of equations, Gaussian elimination or Cramer's rule. Answer the question from the perspective of time and space complexities.
gpt
Both Gaussian elimination and Cramer's rule are methods used to solve systems of linear equations, but they have different time and space complexities which can make one more efficient than the other in certain situations.

Gaussian Elimination:
Gaussian elimination generally involves three steps: forward elimination, back substitution, and normalization. The time complexity of Gaussian elimination is O(n^3) for an n x n system of equations. This is because each elimination operation, which must be performed n times for each of n rows of the matrix, involves O(n) operations. The space complexity is O(n^2) because we need to store the entire matrix.

Cramer's Rule:
Cramer's rule involves computing determinants. To solve an n x n system of equations using Cramer's rule, you must calculate n+1 determinants: one for the coefficient matrix and one for each of the matrices obtained by replacing one column of the coefficient matrix with the right-hand side of the equations. Determinant computation has a time complexity of O(n!) by expansion of minors, or O(n^3) using Gaussian elimination. Therefore, the overall time complexity of Cramer's rule is O(n^4) or O(n^4) using faster determinant computation methods. The space complexity is O(n^2), as with Gaussian elimination.

Given these complexities, Gaussian elimination is generally faster and thus more efficient for larger systems of equations. Cramer's rule, while it can be more straightforward and involve fewer steps for small systems of equations (n=2 or 3), quickly becomes impractical for larger systems due to its higher time complexity.