Cramer's Rule System Solver

Math Linear Algebra • Determinants and Rank

Written by STEM Calculators Team Published February 22, 2026 Updated February 24, 2026

Solve a square linear system \(Ax=b\) using Cramer’s Rule: \[ x_i=\frac{\det(A_i)}{\det(A)}, \] where \(A_i\) is the matrix formed by replacing column \(i\) of \(A\) with the vector \(b\). Works only when \(\det(A)\neq 0\) (unique solution).

Size (n×n)

Display decimals

Zero tolerance

Steps

Visualize variable

Compare with Gaussian elimination (optional)

Entries accept -3.5, 2e-4, fractions like 7/3, and constants pi, e. The canvas shows how column \(i\) is replaced by \(b\) to form \(A_i\).

Matrix \(A\)

2×2

Vector \(b\)

2×1

System is \(Ax=b\). Cramer’s Rule builds \(A_i\) by replacing column \(i\) of \(A\) with \(b\).

Results

\(\det(A)\)

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Unique solution?

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Solution \(x\) (Cramer)

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Note

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Determinants \(\det(A_i)\)

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Ready

Column replacement visual

Left: \(A\) (highlighted column \(i\)). Right: \(A_i\) (replaced column \(i\) = \(b\)).

Drag to pan • wheel to zoom • double-click to reset view • Play cycles \(i=1..n\)

Enter \(A\) and \(b\), then click “Calculate”.

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Theory

Cramer’s Rule (statement)

Consider a square linear system \(Ax=b\), where \(A\in\mathbb{R}^{n\times n}\), \(x\in\mathbb{R}^{n}\), and \(b\in\mathbb{R}^{n}\). Cramer’s Rule expresses each component of the solution vector \(x\) using determinants: \[ x_i=\frac{\det(A_i)}{\det(A)}. \] Here \(A_i\) is the matrix obtained by replacing column \(i\) of \(A\) with the vector \(b\).

When it works (determinant and uniqueness)

Cramer’s Rule requires \(\det(A)\neq 0\). This condition is equivalent to \(A\) being invertible, which implies the system has a unique solution.

If \(\det(A)=0\), the system may have no solution (inconsistent) or infinitely many solutions (underdetermined inside the column space). Cramer’s Rule does not classify these cases by itself; it only guarantees a unique solution when \(\det(A)\neq 0\).

Why column replacement appears

Determinants are multilinear in the columns of a matrix and change predictably under column operations. If \(A=[a_1\,a_2\,\dots\,a_n]\) is written by columns, then the solution of \(Ax=b\) can be viewed as expressing \(b\) as a linear combination of the columns of \(A\): \[ b=x_1 a_1 + x_2 a_2 + \cdots + x_n a_n. \]

Replacing a column with \(b\) isolates one coefficient via determinant identities. This is the core idea behind the formula \(x_i=\det(A_i)/\det(A)\).

2×2 example (hand computation)

For \(A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\) and \(b=\begin{bmatrix}e\\f\end{bmatrix}\), \[ \det(A)=ad-bc,\quad A_1=\begin{bmatrix}e&b\\f&d\end{bmatrix},\quad A_2=\begin{bmatrix}a&e\\c&f\end{bmatrix}. \] Therefore, \[ x_1=\frac{\det(A_1)}{\det(A)},\qquad x_2=\frac{\det(A_2)}{\det(A)}. \] This matches the inverse formula \(A^{-1}=\frac{1}{\det(A)}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\) when \(\det(A)\neq 0\).

Efficiency and why Gaussian elimination is common

Cramer’s Rule is elegant but becomes inefficient for large \(n\). It requires computing \(n+1\) determinants: one for \(A\) and one for each \(A_i\).

In practice, Gaussian elimination (or LU decomposition) solves large systems much faster and more stably. Elimination reduces \(A\) to an upper-triangular matrix and then uses back-substitution.

This calculator can optionally compute the same solution with Gaussian elimination and compare it to the Cramer result. If both methods agree and the residual \(\|Ax-b\|\) is small, that is a strong numerical sanity check.

Numerical note (tolerance and near-singular matrices)

With floating-point inputs, \(\det(A)\) can be extremely small without being exactly zero due to rounding or noisy data. A very small determinant indicates the system may be ill-conditioned: tiny perturbations in \(A\) or \(b\) can cause large changes in \(x\).

This tool uses a tolerance \(\varepsilon\) to treat values with \(|\det(A)|\le \varepsilon\) as “numerically zero,” which helps avoid misleading outputs when the matrix is effectively singular in computation.

University note: beyond square / pseudoinverse

For non-square systems, Cramer’s Rule does not apply directly because determinants are defined only for square matrices. In data-driven problems, least-squares solutions and the Moore–Penrose pseudoinverse (often computed with SVD) are commonly used instead.

Even so, Cramer’s Rule remains important conceptually: it links systems of equations with determinants, invertibility, and the geometric meaning of volume scaling.

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