82. Cramer's Rule

For a square system where is and invertible (i.e., ), each component of the solution is a ratio of determinants:

where is the matrix with its -th column replaced by .

82.1. Why it works (sketch)

Let be the identity with the -th column replaced by . Then (column-by-column), so

(since is upper triangular with on its -th diagonal entry).

82.2. Practical remarks

Cramer’s rule is theoretically beautiful but computationally awful for :

• Computes determinants of matrices
• Cost: via cofactor expansion, or even with row reduction
• Standard methods (Gaussian elimination, LU decomposition) cost

Use Cramer’s rule for:

• Symbolic solutions (small systems with parameters)
• Theoretical proofs
• Closed-form expressions in or cases

Don’t use it for serious numerical work.

82.3. When it fails

If , the formula divides by zero — meaning is singular and the system either has no solution or infinitely many (depending on ). See Linear System Solutions.

82.4. See also

• Determinant
• Matrix Inverse — Cramer’s rule is essentially with the inverse expanded via cofactors
• Adjugate — alternative inverse formula
• Gaussian Elimination — the practical alternative