Verifique se estas matrizes são inversas:
a) ![A = \left[\begin{array}{ccc}7&3\\5&2\\\end{array}\right]](https://tex.z-dn.net/?f=A+%3D+++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D7%26amp%3B3%5C%5C5%26amp%3B2%5C%5C%5Cend%7Barray%7D%5Cright%5D+ "A = \left[\begin{array}{ccc}7&3\\5&2\\\end{array}\right]")
e ![B = \left[\begin{array}{ccc} -2& \ \ 3\\ \ \ 5& -7\\\end{array}\right]](https://tex.z-dn.net/?f=B+%3D+++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+-2%26amp%3B+%5C+%5C+3%5C%5C+%5C++%5C+5%26amp%3B+-7%5C%5C%5Cend%7Barray%7D%5Cright%5D+ "B = \left[\begin{array}{ccc} -2& \ \ 3\\ \ \ 5& -7\\\end{array}\right]")
b ) ![A = \left[\begin{array}{ccc} \ \ 1&2\\-2&3\\\end{array}\right]](https://tex.z-dn.net/?f=A+%3D+++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+%5C++%5C+1%26amp%3B2%5C%5C-2%26amp%3B3%5C%5C%5Cend%7Barray%7D%5Cright%5D+ "A = \left[\begin{array}{ccc} \ \ 1&2\\-2&3\\\end{array}\right]")
e ![B = \left[\begin{array}{ccc} -3& \ \ 2\\ \ \ 2&-1\\\end{array}\right]](https://tex.z-dn.net/?f=B+%3D+++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+-3%26amp%3B+%5C++%5C+2%5C%5C+%5C++%5C+2%26amp%3B-1%5C%5C%5Cend%7Barray%7D%5Cright%5D+ "B = \left[\begin{array}{ccc} -3& \ \ 2\\ \ \ 2&-1\\\end{array}\right]")
Soluções para a tarefa
Respondido por
1
O produto entre uma matriz e sua inversa deve ser uma matriz identidade (da mesma ordem das matrizes)
________________________
a)
Verificando (fazendo o produto das matrizes):
![A\cdot B=\left[\begin{array}{cc}7&3\\5&2\end{array}\right]\cdot\left[\begin{array}{cc}-2&3\\5&-7\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}(7[-2]+3\cdot5)&(7\cdot3+3[-7])\\(5[-2]+2\cdot5)&(5\cdot3+2[-7])\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}(-14+15)&(21-21)\\(-10+10)&(15-14)\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]](https://tex.z-dn.net/?f=A%5Ccdot+B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D7%26amp%3B3%5C%5C5%26amp%3B2%5Cend%7Barray%7D%5Cright%5D%5Ccdot%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D-2%26amp%3B3%5C%5C5%26amp%3B-7%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%5C%5CA%5Ccdot+B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%287%5B-2%5D%2B3%5Ccdot5%29%26amp%3B%287%5Ccdot3%2B3%5B-7%5D%29%5C%5C%285%5B-2%5D%2B2%5Ccdot5%29%26amp%3B%285%5Ccdot3%2B2%5B-7%5D%29%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%5C%5CA%5Ccdot+B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%28-14%2B15%29%26amp%3B%2821-21%29%5C%5C%28-10%2B10%29%26amp%3B%2815-14%29%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%5C%5CA%5Ccdot+B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%26amp%3B0%5C%5C0%26amp%3B1%5Cend%7Barray%7D%5Cright%5D "A\cdot B=\left[\begin{array}{cc}7&3\\5&2\end{array}\right]\cdot\left[\begin{array}{cc}-2&3\\5&-7\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}(7[-2]+3\cdot5)&(7\cdot3+3[-7])\\(5[-2]+2\cdot5)&(5\cdot3+2[-7])\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}(-14+15)&(21-21)\\(-10+10)&(15-14)\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]")
Como o produto é uma matriz identidade, as matrizes são inversas
b)
![A\cdot B=\left[\begin{array}{cc}1&2\\-2&3\end{array}\right]\cdot\left[\begin{array}{cc}-3&2\\2&-1\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}(1[-3]+2\cdot2)&(1\cdot2+2[-1])\\([-2][-3]+3\cdot2)&([-2]\cdot2+3[-1])\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}(-3+4)&(2-2)\\(6+6)&(-4-3)\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}1&0\\12&-7\end{array}\right]](https://tex.z-dn.net/?f=A%5Ccdot+B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%26amp%3B2%5C%5C-2%26amp%3B3%5Cend%7Barray%7D%5Cright%5D%5Ccdot%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D-3%26amp%3B2%5C%5C2%26amp%3B-1%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%5C%5CA%5Ccdot+B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%281%5B-3%5D%2B2%5Ccdot2%29%26amp%3B%281%5Ccdot2%2B2%5B-1%5D%29%5C%5C%28%5B-2%5D%5B-3%5D%2B3%5Ccdot2%29%26amp%3B%28%5B-2%5D%5Ccdot2%2B3%5B-1%5D%29%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%5C%5CA%5Ccdot+B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%28-3%2B4%29%26amp%3B%282-2%29%5C%5C%286%2B6%29%26amp%3B%28-4-3%29%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%5C%5CA%5Ccdot+B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%26amp%3B0%5C%5C12%26amp%3B-7%5Cend%7Barray%7D%5Cright%5D "A\cdot B=\left[\begin{array}{cc}1&2\\-2&3\end{array}\right]\cdot\left[\begin{array}{cc}-3&2\\2&-1\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}(1[-3]+2\cdot2)&(1\cdot2+2[-1])\\([-2][-3]+3\cdot2)&([-2]\cdot2+3[-1])\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}(-3+4)&(2-2)\\(6+6)&(-4-3)\end{array}\right]\\\\\\A\cdot B=\left[\begin{array}{cc}1&0\\12&-7\end{array}\right]")
Como o produto não é uma matriz identidade, as matrizes não são inversas
________________________
a)
Verificando (fazendo o produto das matrizes):
Como o produto é uma matriz identidade, as matrizes são inversas
b)
Como o produto não é uma matriz identidade, as matrizes não são inversas
Perguntas interessantes