Determinant of $$$\left[\begin{array}{cccc}4 - \lambda & 0 & 1 & 0\\0 & 4 - \lambda & 1 & 0\\1 & 1 & 4 - \lambda & 2\\0 & 0 & 2 & 4 - \lambda\end{array}\right]$$$

Calculate $$$\left|\begin{array}{cccc}4 - \lambda & 0 & 1 & 0\\0 & 4 - \lambda & 1 & 0\\1 & 1 & 4 - \lambda & 2\\0 & 0 & 2 & 4 - \lambda\end{array}\right|$$$.

Subtract column $$$3$$$ multiplied by $$$4 - \lambda$$$ from column $$$1$$$: $$$C_{1} = C_{1} - \left(4 - \lambda\right) C_{3}$$$.

$$$\left|\begin{array}{cccc}4 - \lambda & 0 & 1 & 0\\0 & 4 - \lambda & 1 & 0\\1 & 1 & 4 - \lambda & 2\\0 & 0 & 2 & 4 - \lambda\end{array}\right| = \left|\begin{array}{cccc}0 & 0 & 1 & 0\\\lambda - 4 & 4 - \lambda & 1 & 0\\1 - \left(\lambda - 4\right)^{2} & 1 & 4 - \lambda & 2\\2 \lambda - 8 & 0 & 2 & 4 - \lambda\end{array}\right|$$$

Expand along row $$$1$$$:

$$$\left|\begin{array}{cccc}0 & 0 & 1 & 0\\\lambda - 4 & 4 - \lambda & 1 & 0\\1 - \left(\lambda - 4\right)^{2} & 1 & 4 - \lambda & 2\\2 \lambda - 8 & 0 & 2 & 4 - \lambda\end{array}\right| = \left(0\right) \left(-1\right)^{1 + 1} \left|\begin{array}{ccc}4 - \lambda & 1 & 0\\1 & 4 - \lambda & 2\\0 & 2 & 4 - \lambda\end{array}\right| + \left(0\right) \left(-1\right)^{1 + 2} \left|\begin{array}{ccc}\lambda - 4 & 1 & 0\\1 - \left(\lambda - 4\right)^{2} & 4 - \lambda & 2\\2 \lambda - 8 & 2 & 4 - \lambda\end{array}\right| + \left(1\right) \left(-1\right)^{1 + 3} \left|\begin{array}{ccc}\lambda - 4 & 4 - \lambda & 0\\1 - \left(\lambda - 4\right)^{2} & 1 & 2\\2 \lambda - 8 & 0 & 4 - \lambda\end{array}\right| + \left(0\right) \left(-1\right)^{1 + 4} \left|\begin{array}{ccc}\lambda - 4 & 4 - \lambda & 1\\1 - \left(\lambda - 4\right)^{2} & 1 & 4 - \lambda\\2 \lambda - 8 & 0 & 2\end{array}\right| = \left|\begin{array}{ccc}\lambda - 4 & 4 - \lambda & 0\\1 - \left(\lambda - 4\right)^{2} & 1 & 2\\2 \lambda - 8 & 0 & 4 - \lambda\end{array}\right|$$$

Subtract row $$$2$$$ multiplied by $$$4 - \lambda$$$ from row $$$1$$$: $$$R_{1} = R_{1} - \left(4 - \lambda\right) R_{2}$$$.

$$$\left|\begin{array}{ccc}\lambda - 4 & 4 - \lambda & 0\\1 - \left(\lambda - 4\right)^{2} & 1 & 2\\2 \lambda - 8 & 0 & 4 - \lambda\end{array}\right| = \left|\begin{array}{ccc}- \left(\lambda - 4\right) \left(\lambda^{2} - 8 \lambda + 14\right) & 0 & 2 \lambda - 8\\1 - \left(\lambda - 4\right)^{2} & 1 & 2\\2 \lambda - 8 & 0 & 4 - \lambda\end{array}\right|$$$

Expand along column $$$2$$$:

$$$\left|\begin{array}{ccc}- \left(\lambda - 4\right) \left(\lambda^{2} - 8 \lambda + 14\right) & 0 & 2 \lambda - 8\\1 - \left(\lambda - 4\right)^{2} & 1 & 2\\2 \lambda - 8 & 0 & 4 - \lambda\end{array}\right| = \left(0\right) \left(-1\right)^{1 + 2} \left|\begin{array}{cc}1 - \left(\lambda - 4\right)^{2} & 2\\2 \lambda - 8 & 4 - \lambda\end{array}\right| + \left(1\right) \left(-1\right)^{2 + 2} \left|\begin{array}{cc}- \left(\lambda - 4\right) \left(\lambda^{2} - 8 \lambda + 14\right) & 2 \lambda - 8\\2 \lambda - 8 & 4 - \lambda\end{array}\right| + \left(0\right) \left(-1\right)^{3 + 2} \left|\begin{array}{cc}- \left(\lambda - 4\right) \left(\lambda^{2} - 8 \lambda + 14\right) & 2 \lambda - 8\\1 - \left(\lambda - 4\right)^{2} & 2\end{array}\right| = \left|\begin{array}{cc}- \left(\lambda - 4\right) \left(\lambda^{2} - 8 \lambda + 14\right) & 2 \lambda - 8\\2 \lambda - 8 & 4 - \lambda\end{array}\right|$$$

The determinant of a 2x2 matrix is $$$\left|\begin{array}{cc}a & b\\c & d\end{array}\right| = a d - b c$$$.

$$$\left|\begin{array}{cc}- \left(\lambda - 4\right) \left(\lambda^{2} - 8 \lambda + 14\right) & 2 \lambda - 8\\2 \lambda - 8 & 4 - \lambda\end{array}\right| = \left(- \left(\lambda - 4\right) \left(\lambda^{2} - 8 \lambda + 14\right)\right)\cdot \left(4 - \lambda\right) - \left(2 \lambda - 8\right)\cdot \left(2 \lambda - 8\right) = \lambda^{4} - 16 \lambda^{3} + 90 \lambda^{2} - 208 \lambda + 160$$$

$$$\left|\begin{array}{cccc}4 - \lambda & 0 & 1 & 0\\0 & 4 - \lambda & 1 & 0\\1 & 1 & 4 - \lambda & 2\\0 & 0 & 2 & 4 - \lambda\end{array}\right| = \left(\lambda - 4\right)^{2} \left(\lambda^{2} - 8 \lambda + 10\right) = 16 \left(0.25 \lambda - 1\right)^{2} \left(\lambda^{2} - 8 \lambda + 10\right)$$$A