Adjungirana matrika

Imamo splošno matriko ${\displaystyle 2\times 2\,}$ 

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

Njena adjungirana matrika je

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

Imamo matriko ${\displaystyle A\,}$  z razsežnostjo ${\displaystyle 3\times 3\,}$ 

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

Matrika kofaktorjev

${\displaystyle \mathbf {C} ={\begin{pmatrix}+\left|{\begin{matrix}5&6\\8&9\end{matrix}}\right|&-\left|{\begin{matrix}4&6\\7&9\end{matrix}}\right|&+\left|{\begin{matrix}4&5\\7&8\end{matrix}}\right|\\&&\\-\left|{\begin{matrix}2&3\\8&9\end{matrix}}\right|&+\left|{\begin{matrix}1&3\\7&9\end{matrix}}\right|&-\left|{\begin{matrix}1&2\\7&8\end{matrix}}\right|\\&&\\+\left|{\begin{matrix}2&3\\5&6\end{matrix}}\right|&-\left|{\begin{matrix}1&3\\4&6\end{matrix}}\right|&+\left|{\begin{matrix}1&2\\4&5\end{matrix}}\right|\end{pmatrix}}}$ 

Adjungirano matriko dobimo tako, da zgornjo matriko transponiramo:

${\displaystyle \operatorname {adj} (\mathbf {A} )={\begin{pmatrix}+\left|{\begin{matrix}5&6\\8&9\end{matrix}}\right|&-\left|{\begin{matrix}2&3\\8&9\end{matrix}}\right|&+\left|{\begin{matrix}2&3\\5&6\end{matrix}}\right|\\&&\\-\left|{\begin{matrix}4&6\\7&9\end{matrix}}\right|&+\left|{\begin{matrix}1&3\\7&9\end{matrix}}\right|&-\left|{\begin{matrix}1&3\\4&6\end{matrix}}\right|\\&&\\+\left|{\begin{matrix}4&5\\7&8\end{matrix}}\right|&-\left|{\begin{matrix}1&2\\7&8\end{matrix}}\right|&+\left|{\begin{matrix}1&2\\4&5\end{matrix}}\right|\end{pmatrix}}}$ 

kjer je

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

Za primer numerične matrike vzemimo:

${\displaystyle \operatorname {adj} {\begin{pmatrix}\!-3&\,2&\!-5\\\!-1&\,0&\!-2\\\,3&\!-4&\,1\end{pmatrix}}={\begin{pmatrix}\!-8&18&\!-4\\\!-5&\!12&-1\\\ 4&\!-6&\!2\end{pmatrix}}}$ .

Adjunginane matrike imajo naslednje lastnosti

• Adjungirana matrika enotske matrike je enotska matrika

${\displaystyle \mathrm {adj} (\mathbf {I} )=\mathbf {I} \,}$ 

• Adjungirana matrika zmnožka matrik je enak zmnožku adjungiranih matrik

${\displaystyle \mathrm {adj} (\mathbf {AB} )=\mathrm {adj} (\mathbf {B} )\,\mathrm {adj} (\mathbf {A} )\,}$ 

za vse matrike ${\displaystyle A\,}$  in ${\displaystyle B\,}$ , ki imajo razsežnost ${\displaystyle n\times n\,}$ .

• Adjungiranje ohranja transponiranje:

${\displaystyle \mathrm {adj} (\mathbf {A} ^{T})=\mathrm {adj} (\mathbf {A} )^{T}\,}$ .

• Če je ${\displaystyle \mathbf {A} }$  nesingularna matrika, potem velja tudi

${\displaystyle \det {\big (}\mathrm {adj} (\mathbf {A} ){\big )}=\det(\mathbf {A} )^{n-1}.\,}$ 

Adjugiranje se pojavlja tudi v obrazcih za odvod determinante.

• Lastnosti matrik (angleško)