Let $(A|b)$ be the matrix associated to a system of linear equations with $m$ equations and $n$ unknowns, that is, $A$ has $m$ rows and $n$ columns

• The system is compatible if and only if $\mathrm{rk}(A)=\mathrm{rk}(A|b)$
• The system is determinate if and only if $\mathrm{rk}(A)=\mathrm{rk}(A|b)=n$
• The system is indeterminate if and only if $\mathrm{rk}(A)=\mathrm{rk}(A|b)\lt n$
• The system is incompatible if and only if $\mathrm{rk}(A)\lt \mathrm{rk}(A|b)$