Cramer's Rule¶

Theorem¶

If $\boldsymbol{A}$ is a nonsingular $n\times n$ matrix, then the unique solution of $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ is given by $x_j=\frac{\det(\boldsymbol{B}_j)}{\det(\boldsymbol{A})}$ $\frac{d e t ( B _{j} )}{d e t ( A )}$ , where $\boldsymbol{B}_j$ is obtained from $\boldsymbol{A}$ by replacing column $j$ with $\boldsymbol{b}$ .

Interpretation¶

Cramer's rule expresses each unknown of a square linear system with a nonsingular coefficient matrix as a quotient of two determinants. The denominator is always $\det(\boldsymbol{A})$ ; the numerator is the determinant of $\boldsymbol{A}$ with the column of $x_j$ replaced by the right-hand side $\boldsymbol{b}$ .

Two parallelograms in the plane: one spanned by the columns a1 and a2 with area 5, and a larger one spanned by b and a2 with area 10, whose ratio gives x1 = 2.

For the illustrated system $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ $b$ , $\boldsymbol{A}=(\boldsymbol{a}_1\ \boldsymbol{a}_2)=\begin{pmatrix}3&1\\1&2\end{pmatrix}$ and $\boldsymbol{b}=\begin{pmatrix}7\\4\end{pmatrix}$ $(74)$ . The blue parallelogram spanned by $\boldsymbol{a}_1,\boldsymbol{a}_2$ has area $5$ , the red one spanned by $\boldsymbol{b},\boldsymbol{a}_2$ has area $10$ , so $x_1=\dfrac{\det(\boldsymbol{b},\boldsymbol{a}_2)}{\det(\boldsymbol{a}_1,\boldsymbol{a}_2)}=2$ $2$ ; similarly $x_2=\dfrac{\det(\boldsymbol{a}_1,\boldsymbol{b})}{\det(\boldsymbol{a}_1,\boldsymbol{a}_2)}=\dfrac{5}{5}=1$ $1$ . Replacing $\boldsymbol{a}_1$ by $\boldsymbol{b}=x_1\boldsymbol{a}_1+x_2\boldsymbol{a}_2$ multiplies the area by $x_1$ , while the $\boldsymbol{a}_2$ component only shears it.

Each unknown is a ratio of determinants: $x_j=\det(\boldsymbol{B}_j)/\det(\boldsymbol{A})$ $det (B_{j}) / det (A)$ , where $\boldsymbol{B}_j$ has $\boldsymbol{b}$ in column $j$ . Make $\det(\boldsymbol{A})=0$ and the rule breaks down.

The same argument works in general. The determinant is linear in each column and is zero when two columns are equal, so replacing column $j$ by $\boldsymbol{b}=x_1\boldsymbol{a}_1+\cdots+x_n\boldsymbol{a}_n$ leaves only the term $x_j\det(\boldsymbol{A})$ . The hypothesis $\det(\boldsymbol{A})\neq0$ cannot be dropped: then the system has no solution or infinitely many, so there is no single value of $x_j$ to give, and elimination is needed to decide which case occurs.

Cramer's rule is convenient for $2\times2$ and small $3\times3$ systems, or when only one unknown is needed; an $n\times n$ system needs $n+1$ determinants, while Gauss-Jordan elimination needs one reduction. The rule is the formula $\boldsymbol{x}=\boldsymbol{A}^{-1}\boldsymbol{b}$ written out with the adjugate matrix.

A common slip is to replace a row of $\boldsymbol{A}$ by $\boldsymbol{b}$ . It must be column $j$ , the column holding the coefficients of $x_j$ .

Key Results¶

Cramer's rule applies only when $\boldsymbol{A}$ is nonsingular.
It gives an explicit formula for the unique solution of $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ $b$ .

Examples¶

For a $2\times2$ nonsingular system, $x_1=\det(\boldsymbol{B}_1)/\det(\boldsymbol{A})$ and $x_2=\det(\boldsymbol{B}_2)/\det(\boldsymbol{A})$ $det (B_{2}) / det (A)$ .
If $\det(\boldsymbol{A})=0$ $0$ , Cramer's rule cannot be used.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 4

Aliases: Cramer rule Keywords: determinant formula for linear systems

Last updated: 20 October 2026 03:02