Question

A inversa de $\left[\begin{array}{ccc}y&-3&\\-2&x&\\&&\end{array}\right]$ é a matriz $\left[\begin{array}{ccc}x&x-4&\\x-5&1&\\&&\end{array}\right]$ . Determine x e y.

Answer 1

Temos uma matriz

$\mathbf{A}=\left[\begin{array}{cc} y&-3\\ -2&x \end{array}\right ]$


cuja inversa é a matriz

$\mathbf{A}^{\!\!-1}=\left[\begin{array}{cc} x&x-4\\ x-5&1 \end{array}\right ]$


Sabemos que se multiplicarmos uma matriz pela sua inversa, vamos obter a matriz identidade:

$\mathbf{A}\cdot \mathbf{A}^{\!\!-1}=\mathbf{I}_{2}\\\\\\ \left[\begin{array}{cc} y&-3\\ -2&x \end{array}\right ]\cdot \left[\begin{array}{cc} x&x-4\\ x-5&1 \end{array}\right ]=\left[\begin{array}{cc} 1&0\\ 0&1 \end{array}\right ]\\\\\\\\ \left[\begin{array}{cc} yx-3(x-5)~~&~~y(x-4)-3\cdot 1\\\\ -2x+x(x-5)~~&~~-2(x-4)+x\cdot 1 \end{array} \right ]=\left[\begin{array}{cc} 1&0\\\\ 0&1 \end{array} \right ]\\\\\\\\ \left[\begin{array}{cc} yx-3x+15~~&~~y(x-4)-3\\\\ -2x+x^2-5x~~&~~-2x+8+x \end{array} \right ]=\left[\begin{array}{cc} 1&0\\\\ 0&1 \end{array} \right ]$


$\left[\begin{array}{cc} x(y-3)+15~~&~~y(x-4)-3\\\\ x^2-7x~~&~~-x+8 \end{array} \right ]=\left[\begin{array}{cc} 1&0\\\\ 0&1 \end{array} \right ]$


As matrizes são iguais apenas se todos elementos correspondentes forem iguais entre si.


Da 2ª linha, devemos ter

$\left\{ \!\begin{array}{l} x^2-7x=0\\\\ -x+8=1 \end{array} \right.\\\\\\\\ x=8-1\\\\ \boxed{\begin{array}{c}x=7 \end{array}}$

que satisfaz as duas equações do sistema.


Então, a igualdade entre as matrizes fica

$\left[\begin{array}{cc} 7(y-3)+15~~&~~y(7-4)-3\\\\ 0~~&~~1 \end{array} \right ]=\left[\begin{array}{cc} 1&0\\\\ 0&1 \end{array} \right ]\\\\\\\\ \left[\begin{array}{cc} 7y-21+15~~&~~3y-3\\\\ 0~~&~~1 \end{array} \right ]=\left[\begin{array}{cc} 1&0\\\\ 0&1 \end{array} \right ]\\\\\\\\ \left[\begin{array}{cc} 7y-6~~&~~3y-3\\\\ 0~~&~~1 \end{array} \right ]=\left[\begin{array}{cc} 1&0\\\\ 0&1 \end{array} \right ]$


Da 1ª linha, devemos ter

$\left\{ \!\begin{array}{l} 7y-6=1\\\\ 3y-3=0 \end{array} \right.\\\\\\\\ 7y=1+6\\\\ 7y=7\\\\ \boxed{\begin{array}{c}y=1 \end{array}}$

que satisfaz as duas equações do sistema.

__________

Os valores procurados são

x = 7   e   y = 1


Bons estudos! :-)