Cramer's rule expresses each solution coordinate of A x = b as a determinant ratio. This page visualizes why those ratios appear: replacing one column of A by b changes the signed volume by exactly the corresponding coordinate factor.
The first diagram shows the geometric derivation in R² using signed areas. The second diagram shows the same idea in R³ using signed volumes. The third diagram shows the corresponding determinant-ratio argument in Rⁿ.
In two dimensions, the determinant is signed area. Replacing one column by b isolates one coordinate because the area scales by that coordinate while the remaining direction stays fixed. Click the image to open the full-size version.
In three dimensions, the determinant is 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.
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.
The columns of A form the coordinate directions used to build b. If b = x₁a₁ + x₂a₂ + ... + xₙaₙ, then replacing column k by b produces a determinant in which all terms vanish except the one containing xkak.
Therefore det(Ak) = xk det(A), where Ak is obtained from A by replacing column k with b. Dividing by det(A) gives the Cramer's rule formula.
The R² picture emphasizes signed area and the projection-height interpretation. The R² picture emphasizes signed area and the projection-height interpretation. The 3D picture emphasizes signed volume. The Rⁿ picture emphasizes the same determinant properties algebraically: multilinearity and the fact that a determinant becomes zero when two columns are equal or linearly dependent.
Together they show that Cramer's rule is not just a computational formula, but a determinant-based way to isolate one coordinate at a time.
Related visual: Cramer's Rule — Determinant Ratios Visualized
Related chapter: Cramer's rule
Related work: GraphMath Linear Algebra
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