Find the characteristic polynomial of a $10\times 10$ matrix

Question

Let $$A=\begin{pmatrix} 0 & 1 & ... & & & & & & ...& 0\\ 0 &0 &1 &. .. & & & & & ...&0 \\ 0& 0 &0 &1 &... & & & & ...&0 \\ 0 & 0 & 0 &0 &1 & ... & & & ...&0 \\ .& &... & & & & & & ... & \\ .& & ... & & & &\ddots & & ... & \\ .& &... & & & & & & ... &\\ .& &... & & & & & & ... & \\ 0&0 &0 &... & & & & & ...& 1\\ 10^{10}& 0 & 0 & 0 & 0 &0 & 0& 0 &0 &0 \end{pmatrix}$$

Find the characteristic polynomial. This probably needs a trick I don't know. Does someone know it and want to share it with me?

Answer 1

I will go with 3x3 example cause it's just easier to write it.

I thought it would be a good idea to take the $(A)^t$ but then I am stuck again because i don't know how to calculate the $det(A)^t-tI_{10})$

$(A)^t-tI_3=\begin{pmatrix} -t& 0 & a\\ 1& -t&0 \\ 0& 1 & -t \end{pmatrix}$ so if this was a $10x10$ matrix how should I calculate the determinant ?