Question

sea A la matriz
$\left[\begin{array}{ccc}cos x&senx &0\\-senx&cosx&0\\0&0&1\end{array}\right]$ encuentre A^{-1}[/tex] utilizando el metodo de los cofactores

Answer 1

Cálculo del determinante de A:

Utilizando la última fila, claramente:

        $\mbox{det }A=\cos x\cdot \cos x-(-\sin x)\cdot \sin x=cos^2x+\sin^2 x = 1$

Cálculo de matriz de cofactores:

        $c_{11}=\left|\begin{array}{cc}\cos x&0\\0&1\end{array}\right|=\cos x\\\\ c_{12}=-\left|\begin{array}{cc}-\sin x&0\\0&1\end{array}\right|=\sin x\\\\c_{13}=\left|\begin{array}{cc}-\sin x&\cos x\\0&0\end{array}\right|=0\\\\c_{21}=-\left|\begin{array}{cc}\sin x&0\\0&1\end{array}\right|=-\sin x\\\\c_{22}=\left|\begin{array}{cc}\cos x&0\\0&1\end{array}\right|=\cos x$          $c_{23}=-\left|\begin{array}{cc}\cos x&\sin x\\0&0\end{array}\right|=0\\\\ c_{31}=\left|\begin{array}{cc}\sin x&0\\\cos x&0\end{array}\right|=0\\\\c_{32}=-\left|\begin{array}{cc}\cos x&0\\-\sin x&0\end{array}\right|=0\\\\c_{33}=\left|\begin{array}{cc}\cos x&\sin x\\-\sin x&\cos x\end{array}\right|=1$

La matriz es

        $C=\left(\begin{array}{ccc}\cos x&\sin x&0\\-\sin x&\cos x&0\\0&0&1\end{array}\right)$

Cálculo de la inversa de A:

       $A^{-1}=\frac{C^T}{\mbox{det\,}A}=\boxed{\left(\begin{array}{ccc}\cos x&-\sin x&0\\\sin x&\cos x&0\\0&0&1\end{array}\right)}$