82. Cramer's Rule

For a square system $𝐴\overrightarrow{𝑥}=\overrightarrow{𝑏}$ where $𝐴$ is $𝑛\times 𝑛$ and invertible (i.e., $det\left(𝐴\right)\ne 0$), each component of the solution is a ratio of determinants:

where ${𝐴}_{𝑖}$ is the matrix $𝐴$ with its $𝑖$-th column replaced by $\overrightarrow{𝑏}$.

82.1. Why it works (sketch)

Let ${𝐸}_{𝑖}$ be the identity with the $𝑖$-th column replaced by $\overrightarrow{𝑥}$. Then $𝐴{𝐸}_{𝑖}={𝐴}_{𝑖}$ (column-by-column), so

(since ${𝐸}_{𝑖}$ is upper triangular with ${𝑥}_{𝑖}$ on its $𝑖$-th diagonal entry).

82.2. Practical remarks

Cramer’s rule is theoretically beautiful but computationally awful for $𝑛>3$:

• Computes $𝑛+1$ determinants of $𝑛\times 𝑛$ matrices
• Cost: $𝑂\left(𝑛!\right)$ via cofactor expansion, or $𝑂\left({𝑛}^4\right)$ even with row reduction
• Standard methods (Gaussian elimination, LU decomposition) cost $𝑂\left({𝑛}^3\right)$

Use Cramer’s rule for:

• Symbolic solutions (small systems with parameters)
• Theoretical proofs
• Closed-form expressions in $2\times 2$ or $3\times 3$ cases

Don’t use it for serious numerical work.

82.3. When it fails

If $det\left(𝐴\right)=0$, the formula divides by zero — meaning $𝐴$ is singular and the system either has no solution or infinitely many (depending on $\overrightarrow{𝑏}$). See Linear System Solutions.

82.4. See also

• Determinant
• Matrix Inverse — Cramer’s rule is essentially ${𝐴}^{-1}\overrightarrow{𝑏}$ with the inverse expanded via cofactors
• Adjugate — alternative inverse formula
• Gaussian Elimination — the practical alternative