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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 $\mathrm{rk}(A)\lt \mathrm{rk}(A|b)$