Question

If A = $\left[\begin{array}{cc}4& 2\\ -1& 1\end{array}\right]$, prove that (A − 2I) (A − 3I) = O

Answer 1

$$\begin{gathered} \mathrm{Given}:\left(A-2I\right)\left(A-3I\right) \\ \Rightarrow \left(A-2I\right)\left(A-3I\right)=\left(\left[\begin{array}{cc}4& 2\\ -1& 1\end{array}\right]-2\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\right)\left(\left[\begin{array}{cc}4& 2\\ -1& 1\end{array}\right]-3\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\right) \\ \Rightarrow \left(A-2I\right)\left(A-3I\right)=\left(\left[\begin{array}{cc}4& 2\\ -1& 1\end{array}\right]-\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]\right)\left(\left[\begin{array}{cc}4& 2\\ -1& 1\end{array}\right]-\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]\right) \\ \Rightarrow \left(A-2I\right)\left(A-3I\right)=\left[\begin{array}{cc}4-2& 2-0\\ -1-0& 1-2\end{array}\right]\left[\begin{array}{cc}4-3& 2-0\\ -1-0& 1-3\end{array}\right] \\ \Rightarrow \left(A-2I\right)\left(A-3I\right)=\left[\begin{array}{cc}2& 2\\ -1& -1\end{array}\right]\left[\begin{array}{cc}1& 2\\ -1& -2\end{array}\right] \\ \Rightarrow \left(A-2I\right)\left(A-3I\right)=\left[\begin{array}{cc}2-2& 4-4\\ -1+1& -2+2\end{array}\right] \\ \Rightarrow \left(A-2I\right)\left(A-3I\right)=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \Rightarrow \left(A-2I\right)\left(A-3I\right)=0 \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$