If $A=\left[\begin{array}{lll}1 & 0 & 0 \ x & 1 & 0 \ x & x & 1\end{array} ight]$ and $I=\left[\b

Question

If $A=\left[\begin{array}{lll}1 & 0 & 0 \ x & 1 & 0 \ x & x & 1\end{array}ight]$ and $I=\left[\begin{array}{lll}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}ight]$, then $A^3-3 A^2+3 A+I$ is equal to

Professor PC Thomas Classes · Answer

$(A-I)^3 = 0$. Expanding yields $A^3 - 3A^2 + 3A - I = 0 \Rightarrow A^3 - 3A^2 + 3A + I = 2I$.