ماتریس الحاقی

ماتریس الحاقی ماتریس ۲ × ۲

${\displaystyle \mathbf {A} ={\begin{pmatrix}{a}&{b}\\{c}&{d}\end{pmatrix}}}$

برابر است با

${\displaystyle \operatorname {adj} (\mathbf {A} )={\begin{pmatrix}\,\,\,{d}&\!\!{-b}\\{-c}&{a}\end{pmatrix}}.}$

ماتریس ۳ × ۳ زیر را در نظر بگیرید

${\displaystyle \mathbf {A} ={\begin{pmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}}={\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}}.}$

ماتریس الحاقی، ترانهادهٔ ماتریس همسازهٔ آن است، پس

${\displaystyle \mathbf {C} ={\begin{pmatrix}+\left|{\begin{matrix}A_{22}&A_{23}\\A_{32}&A_{33}\end{matrix}}\right|&-\left|{\begin{matrix}A_{21}&A_{23}\\A_{31}&A_{33}\end{matrix}}\right|&+\left|{\begin{matrix}A_{21}&A_{22}\\A_{31}&A_{32}\end{matrix}}\right|\\&&\\-\left|{\begin{matrix}A_{12}&A_{13}\\A_{32}&A_{33}\end{matrix}}\right|&+\left|{\begin{matrix}A_{11}&A_{13}\\A_{31}&A_{33}\end{matrix}}\right|&-\left|{\begin{matrix}A_{11}&A_{12}\\A_{31}&A_{32}\end{matrix}}\right|\\&&\\+\left|{\begin{matrix}A_{12}&A_{13}\\A_{22}&A_{23}\end{matrix}}\right|&-\left|{\begin{matrix}A_{11}&A_{13}\\A_{21}&A_{23}\end{matrix}}\right|&+\left|{\begin{matrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{matrix}}\right|\end{pmatrix}}={\begin{pmatrix}+\left|{\begin{matrix}5&6\\8&9\end{matrix}}\right|&-\left|{\begin{matrix}4&6\\7&9\end{matrix}}\right|&+\left|{\begin{matrix}4&5\\7&8\end{matrix}}\right|\\&&\\-\left|{\begin{matrix}2&3\\8&9\end{matrix}}\right|&+\left|{\begin{matrix}1&3\\7&9\end{matrix}}\right|&-\left|{\begin{matrix}1&2\\7&8\end{matrix}}\right|\\&&\\+\left|{\begin{matrix}2&3\\5&6\end{matrix}}\right|&-\left|{\begin{matrix}1&3\\4&6\end{matrix}}\right|&+\left|{\begin{matrix}1&2\\4&5\end{matrix}}\right|\end{pmatrix}}.}$

بنابراین خواهیم داشت

${\displaystyle \operatorname {adj} (\mathbf {A} )={\begin{pmatrix}+\left|{\begin{matrix}A_{22}&A_{23}\\A_{32}&A_{33}\end{matrix}}\right|&-\left|{\begin{matrix}A_{12}&A_{13}\\A_{32}&A_{33}\end{matrix}}\right|&+\left|{\begin{matrix}A_{12}&A_{13}\\A_{22}&A_{32}\end{matrix}}\right|\\&&\\-\left|{\begin{matrix}A_{21}&A_{23}\\A_{31}&A_{33}\end{matrix}}\right|&+\left|{\begin{matrix}A_{11}&A_{13}\\A_{31}&A_{33}\end{matrix}}\right|&-\left|{\begin{matrix}A_{11}&A_{13}\\A_{21}&A_{23}\end{matrix}}\right|\\&&\\+\left|{\begin{matrix}A_{21}&A_{22}\\A_{31}&A_{32}\end{matrix}}\right|&-\left|{\begin{matrix}A_{11}&A_{12}\\A_{31}&A_{32}\end{matrix}}\right|&+\left|{\begin{matrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{matrix}}\right|\end{pmatrix}}={\begin{pmatrix}+\left|{\begin{matrix}5&6\\8&9\end{matrix}}\right|&-\left|{\begin{matrix}2&3\\8&9\end{matrix}}\right|&+\left|{\begin{matrix}2&3\\5&6\end{matrix}}\right|\\&&\\-\left|{\begin{matrix}4&6\\7&9\end{matrix}}\right|&+\left|{\begin{matrix}1&3\\7&9\end{matrix}}\right|&-\left|{\begin{matrix}1&3\\4&6\end{matrix}}\right|\\&&\\+\left|{\begin{matrix}4&5\\7&8\end{matrix}}\right|&-\left|{\begin{matrix}1&2\\7&8\end{matrix}}\right|&+\left|{\begin{matrix}1&2\\4&5\end{matrix}}\right|\end{pmatrix}}}$

که

${\displaystyle \left|{\begin{matrix}A_{im}&A_{in}\\\,\,A_{jm}&A_{jn}\end{matrix}}\right|=\det \left({\begin{matrix}A_{im}&A_{in}\\\,\,A_{jm}&A_{jn}\end{matrix}}\right).}$