[tex]\mathbf A=\begin{bmatrix}4&0&0\\1&3&0\\-4&5&-1\end{bmatrix}[/tex]
The characteristic polynomial is given by [tex]\det(\mathbf A-\lambda\mathbf I)[/tex]:
[tex]\mathbf A-\lambda\mathbf I=\begin{bmatrix}4-\lambda&0&0\\1&3-\lambda&0\\-4&5&-1-\lambda\end{bmatrix}[/tex]
Compute the determinant by Laplace expansion along the first row:
[tex]\det(\mathbf A-\lambda\mathbf I)=(4-\lambda)\begin{vmatrix}3-\lambda&0\\5&-1-\lambda\end{vmatrix}=(4-\lambda)(3-\lambda)(-1-\lambda)[/tex]
[tex]\implies\det(\mathbf A-\lambda\mathbf I)=-\lambda^3+6\lambda^2-5\lambda-12[/tex]
