Frequently Asked Questions
When does Cramer's rule work?
Whenever the coefficient determinant det(A) is nonzero. If det(A) is zero, the system has no unique solution and Cramer's rule does not apply.
Is Cramer's rule efficient?
For small systems (2x2 or 3x3), yes. For larger systems it is impractical: cofactor expansion grows as n factorial. Gauss-Jordan elimination runs in n cubed.
Does Cramer's rule work for non-square systems?
No. It requires a square invertible coefficient matrix. For non-square systems, use the pseudoinverse or least-squares methods.
How do you use Cramer's rule for a 3x3 system?
Compute the determinant of the coefficient matrix, det(A). Then for each variable, replace the matching column with the constants, take that determinant, and divide by det(A). Each variable equals its replaced-column determinant over det(A).
When does Cramer's rule fail?
It fails when det(A) = 0, because you cannot divide by zero. That happens when the system has either no solution or infinitely many, so you fall back to elimination or the matrix inverse to diagnose which case it is.
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