Question

If A = $\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right]$, show that A² = O

Answer 1

$$\begin{gathered} \mathrm{Given}:A=\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right] \\ \mathrm{Now}, \\ {A}^2=AA \\ \Rightarrow A^2=\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right]\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}{a}^2{b}^2-a^2{b}^2& a{b}^3-a{b}^3\\ -a^{3}b+a^{3}b& -a^2{b}^2+a^2{b}^2\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \Rightarrow A^2=0 \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$