Question

If $A=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],$ then which one of the following statements is correct

• A. $A^2+I=A^4+I$
• B. $A^2+I=A^4-I$
• C. $A^2+I=-A^4$
• D. $A^2+I=A^4$

Answer 1

The correct option is B $A^2+I=A^4-I$

$A^2=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]=-I\Rightarrow A^4=A^2\cdot A^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I\Rightarrow A^4-I=O\cdots \left(i\right)\Rightarrow A^2+I=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\Rightarrow A^2+I=O\cdots \left(ii\right)$
From $\left(i\right)$ and $\left(ii\right)$
$\Rightarrow A^2+I=A^4-I$