# Find $a,b,c$ so that $\begin{bmatrix} 0 & 1& 0 \\ 0 & 0 & 1\\ a & b & c \end{bmatrix}$ has the characteristic polynomial $-\lambda^3+4\lambda^2+5\lambda+6=0$

Find $a,b,c$ so that $\begin{bmatrix} 0 & 1& 0 \\ 0 & 0 & 1\\ a & b & c \end{bmatrix}$ has the characteristic polynomial $-\lambda^3+4\lambda^2+5\lambda+6=0$

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Lets compute the characteristic equation
$0=\det \begin{bmatrix} -\lambda & 1& 0 \\ 0& -\lambda & 1 \\ a & b & c-\lambda \end{bmatrix}$
$=-\lambda \det \begin{bmatrix} -\lambda & 1 \\ b & c-\lambda \end{bmatrix}-1 \det \begin{bmatrix} 0 & 1 \\ a & c-\lambda \end{bmatrix}$
$=-\lambda [\lambda (\lambda-c)-b]-[-a]=-\lambda^3+c\lambda^2+b\lambda+a$
$=-\lambda^3+4\lambda^2+5\lambda+6.$
Hence
$a=6, b=5, c=4.$