Gọi $A = \left(\matrix{a_{11}&a_{12}&a_{13}\cr a_{21}&a_{22}&a_{23}\cr a_{31}&a_{32}&a_{33}} \right)$
Với $a_{ij}$ là số lẻ $(i = \overline{1,3}; j = \overline{1,3})$
Ta được:
$\quad \det(A) = \left|\matrix{a_{11}&a_{12}&a_{13}\cr a_{21}&a_{22}&a_{23}\cr a_{31}&a_{32}&a_{33}} \right|$
$\Leftrightarrow \det(A) = a_{11}.a_{22}.a_{33} + a_{12}.a_{23}.a_{21} + a_{13}.a_{21}.a_{32} - a_{13}.a_{22}.a_{31} - a_{12}.a_{21}.a_{33} - a_{11}.a_{23}.a_{32}$
Do $a_{ij}$ là số lẻ
nên $a_{11}.a_{22}.a_{33}$ lẻ
$a_{12}.a_{23}.a_{21}$ lẻ
$a_{13}.a_{21}.a_{32}$ lẻ
$a_{13}.a_{22}.a_{31}$ lẻ
$a_{12}.a_{21}.a_{33}$ lẻ
$a_{11}.a_{23}.a_{32}$ lẻ
Ta được:
$a_{11}.a_{22}.a_{33} + a_{12}.a_{23}.a_{21} + a_{13}.a_{21}.a_{32}$ lẻ
$a_{13}.a_{22}.a_{31} + a_{12}.a_{21}.a_{33} + a_{11}.a_{23}.a_{32}$ lẻ
$\Rightarrow a_{11}.a_{22}.a_{33} + a_{12}.a_{23}.a_{21} + a_{13}.a_{21}.a_{32} - a_{13}.a_{22}.a_{31} - a_{12}.a_{21}.a_{33} - a_{11}.a_{23}.a_{32}$ chẵn
Hay $\det(A)$ chẵn
