Why Cramer’s Rule Works: Geometric Derivation in R², R³ and Rⁿ

A concept note on deriving determinant ratios from signed area, signed volume and determinant multilinearity

Cramer's rule expresses each solution coordinate of A x = b as a determinant ratio. The formula is often presented as a computation rule, but it has a direct geometric reason.

In R², the determinant is signed area, so the same determinant ratio can be read as a ratio of signed areas with a common base and corresponding heights. In R³, replacing one column of A by b changes a signed volume by exactly the coordinate factor being isolated. In Rⁿ, the same idea follows from determinant multilinearity and from the fact that a determinant vanishes when two columns become linearly dependent.

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Geometric derivation in R²

In two dimensions, the determinant is signed area. The ratio in Cramer's rule can be seen as a ratio of signed areas with the same base direction and with heights measured along a normal direction. Click the image to open the full-size version.

Geometric derivation of Cramer's rule in R2 using signed areas, projection onto a normal direction and determinant ratios.
In two dimensions, the determinant ratio compares signed areas. After projecting onto a normal direction, the common base cancels and the ratio isolates the selected coordinate.

Geometric derivation in R³

The 3D picture emphasizes signed volume. Replacing one column by b isolates one coordinate because the volume scales by that coordinate while the remaining column directions stay fixed. Click the image to open the full-size version.

Geometric derivation of Cramer's rule in R3 using signed volumes and determinant ratios for the solution coordinates.
In three dimensions, determinant ratios compare signed volumes before and after one column is replaced by b.

Geometric derivation in Rⁿ

The same argument works in any dimension. Expanding b in the column basis of A and using multilinearity of the determinant leaves only the term where b replaces the selected column. Click the image to open the full-size version.

Geometric derivation of Cramer's rule in Rn using determinant multilinearity and column replacement.
In Rⁿ, multilinearity keeps only the term containing the selected coordinate xk.

Concept

Suppose A = [a₁ | a₂ | ... | aₙ] is square and full rank, and A x = b. Then b = x₁a₁ + x₂a₂ + ... + xₙaₙ.

Let Ak be the matrix obtained from A by replacing column k with b. When b is expanded inside the determinant, every term has two proportional columns except the term containing xkak. Therefore det(Ak) = xk det(A).

Key equations

A x = b
A = [a₁ | a₂ | ... | aₙ]
b = x₁a₁ + x₂a₂ + ... + xₙaₙ
Aₖ = A with column k replaced by b
det(Aₖ) = xₖ det(A)
xₖ = det(Aₖ) / det(A)

These equations show why column replacement isolates one solution coordinate at a time. In R² and R³, the same relation can be viewed as signed area or signed volume scaling.

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References

Related concept: GraphMath — Cramer's Rule, Determinant Ratios and 3D Geometry

Related chapter: GraphMath — Cramer's rule