Question

Consider the matrix

Answer 1

$\mathbf A=\begin{bmatrix}4&0&0\\1&3&0\\-4&5&-1\end{bmatrix}$

The characteristic polynomial is given by $\det(\mathbf A-\lambda\mathbf I)$:

$\mathbf A-\lambda\mathbf I=\begin{bmatrix}4-\lambda&0&0\\1&3-\lambda&0\\-4&5&-1-\lambda\end{bmatrix}$

Compute the determinant by Laplace expansion along the first row:

$\det(\mathbf A-\lambda\mathbf I)=(4-\lambda)\begin{vmatrix}3-\lambda&0\\5&-1-\lambda\end{vmatrix}=(4-\lambda)(3-\lambda)(-1-\lambda)$

$\implies\det(\mathbf A-\lambda\mathbf I)=-\lambda^3+6\lambda^2-5\lambda-12$