Find the characteristic polynomial |$\lambda – AI $| for the $5\times 5$ matrix
$$A=\left(\begin{matrix}
0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0\\
0 & 0& 0 & 1 & 0\\
0& 0 & 0 & 0 & 1\\
10^{10} & 0 & 0 & 0 & 0\\
\end{matrix}\right)$$
My attempts:
$$A – \lambda I = \left(\begin{matrix}
-\lambda & 1 & 0 & 0 & 0\\
0 & -\lambda & 1 & 0 & 0\\
0 & 0& -\lambda & 1 & 0\\
0& 0 & 0 & -\lambda & 1\\
10^{10} & 0 & 0 & 0 & -\lambda \\
\end{matrix}\right)$$
From my thinking, the characteristic polynomial |$\lambda – AI $| for the $5\times 5$ matrix is -$\lambda ^5$.
Is it true or false? Any hints/solution will be appreciated. Please help me. Thanks you.
Best Answer
Doing a Laplace expansion along the first column, you get that the determinant of that matrix is equal to\begin{align}-\lambda\begin{vmatrix}-\lambda&1&0&0\\0&-\lambda&1&0\\0&0&-\lambda&1\\0&0&0&-\lambda\end{vmatrix}+10^{10}\begin{vmatrix}1&0&0&0\\-\lambda&1&0&0\\0&-\lambda&1&0\\0&0&-\lambda&1\end{vmatrix}=-\lambda^5+10^{10}.\end{align}