= Cramer's rule
{c}
{title2=$x_i=\det(A_i)/\det(A)$}
{wiki}
For an invertible square <matrix> $A$ and $Ax=b$, replace column $i$ by $b$ to form $A_i$. Multilinearity of the <determinant> and $b=\sum_jx_j A_{\cdot j}$ give $\det A_i=x_i\det A$: all other summands have repeated columns and vanish. Division gives the formula. With integer coefficients this supplies explicit rational solutions and, together with the <Hadamard determinant inequality>, bounds their numerators and denominators.
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