Question

If $A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ satisfies A⁴ = λA, then write the value of λ.

Answer 1

$$\begin{gathered} A^2=A.A \\ \Rightarrow A^2=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}1+1& 1+1\\ 1+1& 1+1\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right] \\ \mathrm{Now}, \\ {A}^4=A^2{A}^2 \\ \Rightarrow A^4=\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right]\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right] \\ \Rightarrow A^4=\left[\begin{array}{cc}4+4& 4+4\\ 4+4& 4+4\end{array}\right] \\ \Rightarrow A^4=\left[\begin{array}{cc}8& 8\\ 8& 8\end{array}\right] \\ \mathrm{Also}, \\ {A}^4=\lambda A \\ \Rightarrow \left[\begin{array}{cc}8& 8\\ 8& 8\end{array}\right]=\lambda \left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}8& 8\\ 8& 8\end{array}\right]=\left[\begin{array}{cc}\lambda & \lambda \\ \lambda & \lambda \end{array}\right] \\ \therefore \lambda =8 \end{gathered}$$