## Compute the characteristic polynomial of a companion matrix

Let be a monic polynomial (i.e. ) and let where if , 1 if , and 0 otherwise. Show that .

Recall that , where if and 0 otherwise.

Using the Cofactor Expansion Formula along the th column (derived here), we have , where denotes the matrix minor of .

Evidently, we have that if , if , if and , if and , and 0 otherwise. So , where is lower triangular of dimension and diagonal entries equal to and is upper triangular of dimension and diagonal entries equal to . So we have .

So we have as desired.

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## Comments

This proof is incorrect, since $M_{k,n}$ is clearly not upper triangular for $k>1$. Instead it is block-triangular (i.e., of 2×2 block form with the lower left block null and touching the diagonal) with the top-left block, square of size $k-1$, being LOWER triangular with $x$’s on the diagonal, and the bottom right block of siwe $n-k$ being upper triangular with $-1$’s on the diagonal.

Thanks! I think it is fixed now.