Cramer's Rule

Precise definition

For a square system $A\mathbf x=\mathbf b$ with $\det A\ne0$, Cramer's rule gives $x_i=\det A_i/\det A$, where $A_i$ is obtained by replacing column $i$ of $A$ by $\mathbf b$. The nonzero determinant is essential: it means the system has a unique solution.

Notation and mathematical language

Column replacement matches the position of the unknown because $A\mathbf x$ is a linear combination of the columns of $A$. Cramer's rule applies to $n$ equations in $n$ unknowns. It is a theoretical identity and can be convenient for tiny symbolic systems.

Conceptual picture

Multilinearity and alternation of the determinant isolate one coefficient when a column is replaced by the right-hand side. Every other term repeats a column and has determinant zero; the surviving term is $x_i\det A$.

Conditions and key results

If $\det A=0$, the ratios are undefined and the system may have no solutions or infinitely many; Cramer's rule cannot decide which. For large numerical systems it is inefficient and often less stable than elimination or factorization.

A reliable strategy

1. Confirm the coefficient matrix is square and compute $\det A$.
2. If the determinant is nonzero, create $A_i$ by replacing the correct column—not a row—with $\mathbf b$.
3. Compute each determinant with signs and form $x_i=\det A_i/\det A$.
4. Substitute the vector into every original equation; choose elimination instead when size or conditioning makes it preferable.

Fully worked example

Interpretation and application

Cramer's rule shows each solution component depends rationally on entries of $A$ and $\mathbf b$. This is useful in proofs and sensitivity formulas, but near-zero determinants can magnify perturbations; determinant size alone is not a scale-invariant condition number.

Common mistakes

Verification and reasonableness

• Multiply $A\mathbf x$ and recover $\mathbf b$.
• Compare the result with one pass of Gaussian elimination.
• Recompute determinant signs by a second expansion or row-operation check.

Practice

1. What condition is required for Cramer's rule?
2. Which part of $A$ is replaced to find $x_i$?
3. What does $\det A=0$ imply about the rule?

Further deduction

The rule follows from the adjugate identity $A\operatorname{adj}(A)=(\det A)I$. When $\det A\ne0$, $A^{-1}=\operatorname{adj}(A)/\det A$, and multiplying by $\mathbf b$ produces the determinant ratios. This connection also explains why the same invertibility hypothesis appears in Cramer's rule and the inverse formula.

For parameterized symbolic systems, the determinant condition identifies exceptional parameter values before division. If $A(k)$ has $\det A(k)=0$, solve those values separately by row reduction; substituting them into determinant ratios is invalid. This case split often reveals a unique solution for generic $k$ and none or infinitely many at special values.

Geometrically in two dimensions, $\det A$ is signed area scaling. Replacing one column by $b$ compares the area needed for the corresponding coordinate, which gives the determinant ratio. A zero base area explains both division failure and loss of unique coordinates.

Related topics

• Determinants
• Gaussian Elimination
• Inverse Matrix