Given the matrix $A=\left[\begin{array}{ccc}0 & 1 & 0 \ 0 & 0 & 1 \ 1 & 2 & -1\end{array} ight]$.

Question

Given the matrix $A=\left[\begin{array}{ccc}0 & 1 & 0 \ 0 & 0 & 1 \ 1 & 2 & -1\end{array}ight]$. If constants $p, q, r$ satisfy $A^3=p A^2+q A+r I$, then

Professor PC Thomas Classes · Answer

Apply Cayley-Hamilton theorem. Characteristic equation evaluates coefficients to $p=-1, q=2, r=1$.