Cramer's Rule for Solving Linear Systems

What is Cramer's rule?

For a square linear system $A \mathbf{x} = \mathbf{b}$ with $\det A \ne 0$, each variable is a ratio of determinants: $x_i = \dfrac{\det(A_i)}{\det(A)}$

where $A_i$ is the matrix obtained by replacing column $i$ of $A$ with the right-hand side $\mathbf{b}$.

Step-by-step

Setup: $A = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 5 \\ 7 \end{pmatrix}$.

Step 1 — $\det A$: $\det A = (3)(4) - (2)(1) = 12 - 2 = 10$

Since $\det A \ne 0$, Cramer's rule applies.

Step 2 — $A_1$: replace column 1 with $\mathbf{b}$. $A_1 = \begin{pmatrix} 5 & 2 \\ 7 & 4 \end{pmatrix}, \ \det A_1 = (5)(4) - (2)(7) = 20 - 14 = 6$

$x = \dfrac{\det A_1}{\det A} = \dfrac{6}{10} = \boxed{\dfrac{3}{5}}$

Step 3 — $A_2$: replace column 2 with $\mathbf{b}$. $A_2 = \begin{pmatrix} 3 & 5 \\ 1 & 7 \end{pmatrix}, \ \det A_2 = (3)(7) - (5)(1) = 21 - 5 = 16$

$y = \dfrac{\det A_2}{\det A} = \dfrac{16}{10} = \boxed{\dfrac{8}{5}}$

Verification

• Eq 1: $3(3/5) + 2(8/5) = 9/5 + 16/5 = 25/5 = 5$ ✓
• Eq 2: $(3/5) + 4(8/5) = 3/5 + 32/5 = 35/5 = 7$ ✓

Why it works

From $A \mathbf{x} = \mathbf{b}$, replacing column $i$ of $A$ with $\mathbf{b}$ gives a matrix whose determinant expands as $x_i \det A$ (the other $x_k$ terms vanish because they'd create repeated columns). So $\det(A_i) = x_i \det(A)$, yielding Cramer's formula.

When to use it

• Small systems ($2 \times 2$, $3 \times 3$). Beautiful closed form, easy to verify.
• Symbolic work with few unknowns where you want a clean expression in terms of the coefficients.
• Theoretical arguments about how solutions depend on parameters.

Not the fastest computational method for large systems. Gaussian elimination runs in $O(n^3)$; Cramer's rule via cofactor expansion is $O(n!)$ — exponential. For $n \ge 4$, prefer row reduction or matrix inversion.

When it FAILS

If $\det A = 0$, the system is either inconsistent (no solution) or dependent (infinitely many solutions). Cramer's rule doesn't distinguish between these — you need to fall back to row reduction to diagnose which case you're in.

Common mistakes

• Replacing the wrong column. $A_i$ uses column $i$, not row $i$.
• Skipping the determinant check. Always confirm $\det A \ne 0$ first.
• Wrong sign on cofactor sub-determinants (in $3 \times 3$ or larger cases).

Try it in the visualization

Slide $\mathbf{b}$ and watch $x, y$ update as ratios. The three matrices $A, A_1, A_2$ are shown side by side, with the replaced columns highlighted in gold.