Applying Cramer's Rule

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Applying Cramer's Rule Solve a 3×3 linear system by replacing columns and computing determinants Use Cramer's Rule to solve the linear system below for x , y , and z . In matrix form , with , each unknown is , where A_i is A with its i -th column replaced by . Nicely done. You solved a 3×3 system end to end with Cramer's Rule and confirmed the answer. The formula: for , every unknown is a single ratio . Column replacement: A_i is the coefficient matrix with its i -th column swapped for the constants — only that one column changes. Why it works here: , so the system has a unique solution; if the rule does not apply. The result: , so , and (2,2,2) satisfies every equation. Cramer's Rule turns "solve the system" into "compute a few determinants" — elegant for small systems, though Gaussian elimination scales better for large ones.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.