Question
Find the characteristic polynomial of the matrix
$$ A = \left( \begin{matrix}12&-4\\-3&1\end{matrix} \right) $$Answer
The characteristic polynomial for matrix A is:
$$ p(\lambda) = \lambda^2-13\lambda $$Explanation
The characteristic polynomial is given by
$$ p(\lambda) = det(A - \lambda I) $$In this example we have:
$$ \begin{aligned} p(\lambda) &= det(A - \lambda I) = det \left( \left[ \begin{matrix}12&-4\\-3&1\end{matrix} \right] - \left[ \begin{matrix} \lambda & 0 \\ 0 & \lambda \end{matrix} \right] \right) = \\ &= \left| \begin{matrix}12 - \lambda &-4\\-3&1 - \lambda \end{matrix} \right| = (-\lambda+12) (-\lambda+1) - (-4) \cdot (-3) = \\ &= \lambda^2-13\lambda+12 - 12 = \lambda^2-13\lambda \end{aligned} $$This page was created using
Characteristic polynomial calculator