Cramer's rule ($x_i=\det (A_i)/\det (A)$)

For an invertible square matrix $A$ and $Ax=b$, replace column $i$ by $b$ to form $A_i$. Multilinearity of the determinant and $b={\sum }_j{x}_j{A}_{\cdot j}$ 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.