Question

If f (x) = x² − 2x, find f (A), where $A=\left[\begin{array}{ccc}0& 1& 2\\ 4& 5& 0\\ 0& 2& 3\end{array}\right]$

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}f\left(x\right)=x^2-2x \\ f\left(A\right)=A^2-2A \\ \mathrm{Now}, \\ {A}^2=AA \\ \Rightarrow A^2=\left[\begin{array}{ccc}0& 1& 2\\ 4& 5& 0\\ 0& 2& 3\end{array}\right]\left[\begin{array}{ccc}0& 1& 2\\ 4& 5& 0\\ 0& 2& 3\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{ccc}0+4+0& 0+5+4& 0+0+6\\ 0+20+0& 4+25+0& 8+0+0\\ 0+8+0& 0+10+6& 0+0+9\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{ccc}4& 9& 6\\ 20& 29& 8\\ 8& 16& 9\end{array}\right] \\ f\left(A\right)=A^2-2A \\ \Rightarrow f\left(A\right)=\left[\begin{array}{ccc}4& 9& 6\\ 20& 29& 8\\ 8& 16& 9\end{array}\right]-2\left[\begin{array}{ccc}0& 1& 2\\ 4& 5& 0\\ 0& 2& 3\end{array}\right] \\ \Rightarrow f\left(A\right)=\left[\begin{array}{ccc}4& 9& 6\\ 20& 29& 8\\ 8& 16& 9\end{array}\right]-\left[\begin{array}{ccc}0& 2& 4\\ 8& 10& 0\\ 0& 4& 6\end{array}\right] \\ \Rightarrow f\left(A\right)=\left[\begin{array}{ccc}4-0& 9-2& 6-4\\ 20-8& 29-10& 8-0\\ 8-0& 16-4& 9-6\end{array}\right] \\ \Rightarrow f\left(A\right)=\left[\begin{array}{ccc}4& 7& 2\\ 12& 19& 8\\ 8& 12& 3\end{array}\right] \end{gathered}$$