Proposition 13

For a system of $n$ linear equations in $n$ unknowns over a commutative ring to admit at least one solution regardless of the right hand sides, it is necessary and sufficient that the determinant of the matrix of the system be invertible; in this case the system admits a single solution.

If $\det L$ is not a divisor of zero in $\mathbf{A}$, equation (34) is equivalent to the equation

If $M$ is the cofactor matrix of $L$, we derive from (34) and formula (26) of no. 6 the relation (36) which may also be written (37) where $L_i$ denotes the matrix obtained by replacing the column of $L$ of index $i$ by $y$. Formulae (37) are called Cramer's formulae for system (34); every solution of (34) is also a solution of (37). Conversely, we derive from (36), taking account of formula (28) of no. 6, (38) and hence, if $\det L$ is not a divisor of zero in $\mathbf{A}$, systems (34) and (37) are equivalent; if $\det L$ is invertible, the unique solution of (34) is given by (39) A system (34) such that $\det L$ is invertible is also called a Cramer system.

In particular let $y = 0$; it then follows from Proposition 3 of no. 2 that: