Considere a matriz A= (aij)3x3 em que:
x, se i = j
1, se i ≠ j
Determine os valores de x para os quais Det A = 2
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Escrevendo a matriz genérica, temos;
![A=\left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right]](https://tex.z-dn.net/?f=A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Da_%7B11%7D%26amp%3Ba_%7B12%7D%26amp%3Ba_%7B13%7D%5C%5Ca_%7B21%7D%26amp%3Ba_%7B22%7D%26amp%3Ba_%7B23%7D%5C%5Ca_%7B31%7D%26amp%3Ba_%7B32%7D%26amp%3Ba_%7B33%7D%5Cend%7Barray%7D%5Cright%5D+ "A=\left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right]")
Então,
![A= \left[\begin{array}{ccc}x&1&1\\1&x&1\\1&1&x\end{array}\right] \\\\\\\left[\begin{array}{ccc}x&1&1\\1&x&1\\1&1&x\end{array}\right] =2\\\\\\x^3+1+1-x-x-x=2\\\\x^3-3x=0\\\\x(x^2-3)=0\\\\x_1=0\\\\x_2= \sqrt{3}\\\\x_3=-\sqrt{3}](https://tex.z-dn.net/?f=A%3D+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%26amp%3B1%26amp%3B1%5C%5C1%26amp%3Bx%26amp%3B1%5C%5C1%26amp%3B1%26amp%3Bx%5Cend%7Barray%7D%5Cright%5D+%5C%5C%5C%5C%5C%5C%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%26amp%3B1%26amp%3B1%5C%5C1%26amp%3Bx%26amp%3B1%5C%5C1%26amp%3B1%26amp%3Bx%5Cend%7Barray%7D%5Cright%5D+%3D2%5C%5C%5C%5C%5C%5Cx%5E3%2B1%2B1-x-x-x%3D2%5C%5C%5C%5Cx%5E3-3x%3D0%5C%5C%5C%5Cx%28x%5E2-3%29%3D0%5C%5C%5C%5Cx_1%3D0%5C%5C%5C%5Cx_2%3D+%5Csqrt%7B3%7D%5C%5C%5C%5Cx_3%3D-%5Csqrt%7B3%7D "A= \left[\begin{array}{ccc}x&1&1\\1&x&1\\1&1&x\end{array}\right] \\\\\\\left[\begin{array}{ccc}x&1&1\\1&x&1\\1&1&x\end{array}\right] =2\\\\\\x^3+1+1-x-x-x=2\\\\x^3-3x=0\\\\x(x^2-3)=0\\\\x_1=0\\\\x_2= \sqrt{3}\\\\x_3=-\sqrt{3}")
Espero ter ajudado
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Então,
Espero ter ajudado
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