Cramer's rule

Statement

Let $F$ be a field, let $A$ be an $n \times n$ matrix over $F$ with $\det A \ne 0$, and let $b \in F^n$. For $1 \le i \le n$, let $A_i$ be the matrix that results when column $i$ of $A$ is replaced by $b$. Then the linear system $Ax = b$ has exactly one solution $x = (x_1, \dots, x_n)$, and $$x_i = \frac{\det A_i}{\det A} \qquad (1 \le i \le n).$$

Remarks

For $n = 2$, the system $ax + by = e$, $cx + dy = f$ has the solution $x = (ed - bf)/(ad - bc)$ and $y = (af - ec)/(ad - bc)$. The rule follows from the formula for the inverse with the adjugate. It needs $n + 1$ determinants, so it is slow for large $n$ compared with Gaussian elimination.

Explanations

L1 human

Cramer's rule gives each unknown as a quotient of two determinants. The denominator is always $\det A$. For the numerator of unknown $i$, put the right side $b$ in column $i$ of $A$.