Для матрицы найдите все значения x ∈ C, при которых матрица A − xE необратима.

$
A = \left(
\begin{array}{cc}
7 + 6i & 2 + i
\\
-12 - 6i & -3 + i
\\
\end{array}
\right)
\\\\\\
B = A - xE = \left(
\begin{array}{cc}
7 + 6i - a - bi & 2 + i
\\
-12 - 6i & -3 + i - a - bi
\\
\end{array}
\right), x = a + bi
\\\\
det(B) = 0 => B \text{ необратима}
\\\\
|B| = \left|
\begin{array}{cc}
7 + 6i - a - bi & 2 + i
\\
-12 - 6i & -3 + i - a - bi
\\
\end{array}
\right|
= (-3 + i - a - bi)(7 + 6i - a - bi) - (2 + i)(-12 - 6i) = -27 - 11i - 4a - 7ai -4bi + 7b + a^2 + 2abi - b^2 + 18 + 24i = -9 + 13i - 4a - 7ai - 4bi + 7b + a^2 + 2abi - b^2 = (a - 3 + (b - 4)i)(a - 1 + (b - 3)i) = 0
\\\\
1) a - 3 + (b - 4)i = 0 => a = 3, b = 4
\\\\
2) a - 1 + (b - 3)i = 0 => a = 1, b = 3
\\\\
\text{Ответ: } x = 3 + 4i, x = 1 + 3i
$