Inversión Matriz de Cofactores

Sea la Matriz:

$$A=\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}$$

La matriz de cofactores es:

$$cof(A)=\begin{bmatrix}c_{11}&c_{12}&c_{13} \\ c_{21}&c_{22}&c_{23} \\ c_{31}&c_{32}&c_{33}\end{bmatrix} \qquad |A_{11}|=\begin{vmatrix}e&f \\ h&i\end{vmatrix}$$

$$c_{11}=(-1)^{1+1}|A_{11}| \qquad c_{12}=(-1)^{1+2}|A_{12}|$$

$$c_{13}=(-1)^{1+3}|A_{13}| \qquad c_{21}=(-1)^{2+1}|A_{21}|$$

$$c_{22}=(-1)^{2+2}|A_{22}| \qquad c_{23}=(-1)^{2+3}|A_{23}|$$

$$c_{31}=(-1)^{3+1}|A_{31}| \qquad c_{32}=(-1)^{3+2}|A_{32}|$$

$$c_{33}=(-1)^{3+3}|A_{33}|$$

$$cof(A)=\begin{bmatrix}+\begin{vmatrix}e&f\\h&i\end{vmatrix}&-\begin{vmatrix}d&f\\g&i\end{vmatrix}&+\begin{vmatrix}d&e\\g&h \end{vmatrix} \\-\begin{vmatrix}b&c\\h&i\end{vmatrix}&+\begin{vmatrix}a&c\\g&i\end{vmatrix}&-\begin{vmatrix}a&b\\g&h\end{vmatrix} \\+\begin{vmatrix}b&c\\e&f\end{vmatrix}&-\begin{vmatrix}a&c\\d&f\end{vmatrix}&+\begin{vmatrix}a&b\\d&e\end{vmatrix}\end{bmatrix}$$

$$A^{-1}=\frac{1}{|A|}adj(A)=\frac{1}{|A|}[cof(A)]^t$$