Question

Calcule a matriz inversa, se existir das matrizes abaixo:

Answer 1

$\mathrm{\mathbf{a)}\ A=\left[\begin{array}{cc}7&-3\\2&1\end{array}\right]\ \to\ \det{A}=7.1-(-3).2=7+6=13}\\\\\\ \mathrm{A^{-1}=\dfrac{1}{\det{A}}\left[\begin{array}{cc}cof(a_{11})&cof(a_{12})\\cof(a_{21})&cof(a_{22})\end{array}\right]^t}\\\\\\ \mathrm{A^{-1}=\dfrac{1}{13}\left[\begin{array}{cc}1&-2\\3&7\end{array}\right]^t\ \to\ A^{-1}=\dfrac{1}{13}\left[\begin{array}{cc}1&3\\-2&7\end{array}\right]}\\\\\\ \boxed{\mathrm{A^{-1}=\left[\begin{array}{cc}1/13&3/13\\-2/13&7/13\end{array}\right]}}$

$\mathrm{\mathbf{b)}\ B=\left[\begin{array}{ccc}1&2&3\\0&1&4\\0&0&1\end{array}\right]\ \to\ \det{B}=1.1.1=1}\\\\\\ \mathrm{B^{-1}=\dfrac{1}{\det{B}}\left[\begin{array}{ccc}cof(a_{11})&cof(a_{12})&cof(a_{13})\\cof(a_{21})&cof(a_{22})&cof(a_{23})\\cof(a_{31})&cof(a_{32})&cof(a_{33})\end{array}\right]^t}\\\\\\ \mathrm{B^{-1}=\dfrac{1}{1}\left[\begin{array}{ccc}1&0&0\\-2&1&0\\5&-4&1\end{array}\right]^t\ \to\ \boxed{\mathrm{B^{-1}=\left[\begin{array}{ccc}1&-2&5\\0&1&-4\\0&0&1\end{array}\right]}}}$