Adjunkte

$\operatorname {adj} (A)=\operatorname {Cof} (A)^{T}={\tilde {A}}^{T}={\begin{pmatrix}{\tilde {a}}_{11}&{\tilde {a}}_{12}&\cdots &{\tilde {a}}_{1n}\\{\tilde {a}}_{21}&{\tilde {a}}_{22}&&{\tilde {a}}_{2n}\\\vdots &&\ddots &\vdots \\{\tilde {a}}_{n1}&{\tilde {a}}_{n2}&\cdots &{\tilde {a}}_{nn}\end{pmatrix}}^{T}={\begin{pmatrix}{\tilde {a}}_{11}&{\tilde {a}}_{21}&\cdots &{\tilde {a}}_{n1}\\{\tilde {a}}_{12}&{\tilde {a}}_{22}&&{\tilde {a}}_{n2}\\\vdots &&\ddots &\vdots \\{\tilde {a}}_{1n}&{\tilde {a}}_{2n}&\cdots &{\tilde {a}}_{nn}\end{pmatrix}}$.

Es ist hierbei zu beachten, dass an der Stelle $(j,i)$ der Kofaktor ${\tilde {a}}_{ij}$ steht. Die Kofaktoren ${\tilde {a}}_{ij}$ berechnen sich zu

${\tilde {a}}_{ij}=(-1)^{i+j}\cdot M_{ij}=(-1)^{i+j}\cdot \det {\begin{pmatrix}a_{1,1}&\cdots &a_{1,j-1}&a_{1,j+1}&\cdots &a_{1,n}\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\a_{i-1,1}&\cdots &a_{i-1,j-1}&a_{i-1,j+1}&\cdots &a_{i-1,n}\\a_{i+1,1}&\cdots &a_{i+1,j-1}&a_{i+1,j+1}&\cdots &a_{i+1,n}\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\a_{n,1}&\cdots &a_{n,j-1}&a_{n,j+1}&\cdots &a_{n,n}\end{pmatrix}}$.