Cramer's Rule, Volume, and Linear Transformations

The determinant detects invertibility and multiplies over products. Three consequences follow — two closed-form results and one geometric reading:

• Cramer's rule writes each unknown of an invertible system as a ratio of determinants.
• The adjugate formula does the same for the inverse matrix.
• Area and volume: the absolute determinant is the area or volume of the figure spanned by the columns of $A$.

The geometric reading extends furthest. It is the linear special case of the Jacobian factor that governs change of variables in multivariable calculus.

Cramer's rule

For a matrix $A$ and a vector $\mathbf{b}$, write $A_i(\mathbf{b})$ for the matrix obtained by replacing column $i$ of $A$ with $\mathbf{b}$, leaving every other column alone.

The proof uses the multiplicative property. Let $I_i(\mathbf{x})$ be the identity matrix with its $i$th column replaced by $\mathbf{x}$. Matrix multiplication gives $A\cdot I_i(\mathbf{x})=A_i(\mathbf{b})$, so by the multiplicative property, $(\det A)(\det I_i(\mathbf{x}))=\det A_i(\mathbf{b})$. A cofactor expansion of $I_i(\mathbf{x})$ along its $i$th row shows $\det I_i(\mathbf{x})=x_i$, hence $(\det A)x_i=\det A_i(\mathbf{b})$, and dividing by the nonzero $\det A$ finishes the proof.¹

Cramer's rule is efficient only for small systems — each unknown is a fresh determinant — but its value is theoretical. Because $x_i$ is an explicit function of the entries of $A$ and $\mathbf{b}$, the formula shows how a solution responds to changes in the data, which is the question that arises when a coefficient carries a parameter.

The determinant in the denominator names the exact values where the system degenerates. This is the pattern behind Laplace-transform analyses of linear systems in engineering, where $s$ is the transform variable.¹

A formula for the inverse

Applying Cramer's rule column by column produces the inverse in closed form. The $j$th column of $A^{-1}$ solves $A\mathbf{x}={\mathbf{e}}_j$, and the $(i,j)$ entry of $A^{-1}$ is therefore

A cofactor expansion of $A_i({\mathbf{e}}_j)$ down its $i$th column evaluates the numerator to a single cofactor, $\det A_i({\mathbf{e}}_j)=(-1{)}^{i+j}\det A_{ji}=C_{ji}$. Note the reversed subscripts: the $(i,j)$ entry of the inverse uses the $(j,i)$ cofactor of $A$.

Like Cramer's rule, the adjugate formula is a theoretical instrument. It exposes how the inverse depends on the entries of $A$ without computing anything, but for an actual inverse the Gauss-Jordan reduction of $[A\mid I]$ is far cheaper.¹

The determinant as area and volume

The absolute value of the determinant is also a geometric measure.

The proof begins with the easy case: a diagonal matrix $\left[\begin{array}{cc}a& 0\\ 0& d\end{array}\right]$ spans an axis-aligned rectangle of area $|ad|$, matching the determinant. The general case reduces to this one. The absolute determinant is unchanged by a column replacement or a column interchange (the row-operation rules, read for columns), and those operations suffice to diagonalize any $2\times 2$ matrix. Geometrically, a column replacement ${\mathbf{a}}_2\mapsto {\mathbf{a}}_2+c{\mathbf{a}}_1$ slides the tip of ${\mathbf{a}}_2$ along a line parallel to ${\mathbf{a}}_1$, preserving the base and the height of the parallelogram, hence its area.¹

The $3\times 3$ argument is the same in one higher dimension: a diagonal matrix spans a box of volume $|abc|$, and column replacements move one edge within a plane parallel to the opposite face, changing neither the base area nor the height of the parallelepiped.

Translation does not affect area, so any parallelogram can be moved to the origin and read off a determinant this way.

Linear maps scale area and volume

If $T(\mathbf{x})=A\mathbf{x}$ is the linear map with matrix $A$, then applying $T$ to a region multiplies its measure by exactly $|\det A|$.

The proof combines the area-and-volume result with the multiplicative property. A parallelogram at the origin spanned by ${\mathbf{b}}_1,{\mathbf{b}}_2$ has matrix $B=[{\mathbf{b}}_1{\mathbf{b}}_2]$, and its image under $T$ is spanned by $A{\mathbf{b}}_1,A{\mathbf{b}}_2$, with matrix $AB$. Then

An arbitrary parallelogram is a translate $\mathbf{p}+S$, and $T$ sends it to $T(\mathbf{p})+T(S)$; since translation preserves area, the factor $|\det A|$ carries over unchanged.¹

Beyond parallelograms

The scaling factor does not depend on straight edges. A region $R$ with finite area can be approximated by a grid of small squares inside it; the map $T$ sends each square to a parallelogram whose area is $|\det A|$ times the square's.

Summing and passing to a limit shows

for any region $R$, and the analogous statement holds for volume in ${\mathbb{R}}^3$. This is the linear special case of the change-of-variables formula in calculus, where the local scaling factor of a nonlinear map is the absolute value of its Jacobian determinant. The determinant a linear map applies uniformly becomes the rate at which a smooth map expands area near each point.

Summary

• Cramer's rule: for invertible $A$, the solution of $A\mathbf{x}=\mathbf{b}$ is $x_i=\det A_i(\mathbf{b})/\det A$, where $A_i(\mathbf{b})$ replaces column $i$ of $A$ by $\mathbf{b}$.
• The adjugate formula: $A^{-1}=\frac{1}{\det A}\mathrm{adj}A$, where $\mathrm{adj}A$ is the transpose of the cofactor matrix. Both formulas are theoretical tools, not efficient algorithms.
• Area and volume: $|\det A|$ is the area of the parallelogram (or volume of the parallelepiped) spanned by the columns of $A$.
• Measure scaling: a linear map with matrix $A$ scales the area or volume of every region by $|\det A|$, a fact that extends from parallelograms to arbitrary regions and previews the Jacobian in multivariable calculus.