Question

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Answer 1

Explicație pas cu pas:

$A^{2} = {\left(\begin{array}{ccc}0&1&1\\1&0&1\\1&1&0\end{array}\right)}^{2} = \left(\begin{array}{ccc}2&1&1\\1&2&1\\1&1&2\end{array}\right)$

$aA^{2} + bA + 2I_{3} = O_{3}$

$a\left(\begin{array}{ccc}2&1&1\\1&2&1\\1&1&2\end{array}\right) + b\left(\begin{array}{ccc}0&1&1\\1&0&1\\1&1&0\end{array}\right) + 2\left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right) = \left(\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array}\right)$

$\left(\begin{array}{ccc}2a&a&a\\a&2a&a\\a&a&2a\end{array}\right) + \left(\begin{array}{ccc}0&b&b\\b&0&b\\b&b&0\end{array}\right) + \left(\begin{array}{ccc}2&0&0\\0&2&0\\0&0&2\end{array}\right) = \left(\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array}\right)$

$\left(\begin{array}{ccc}2a + 2&a + b&a + b\\a + b&2a + 2&a + b\\a + b&a + b&2a + 2\end{array}\right) = \left(\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array}\right)$

$\begin{cases} 2a + 2 = 0\\ a + b = 0\end{cases} \iff \begin{cases} 2(a + 1) = 0\\ a + b = 0\end{cases}$

$\implies \begin{cases} a = -1 \\ b = 1\end{cases}$

Answer 2

Răspuns:

Explicație pas cu pas: