Mathematics - Matrices Question with Solution | TestHub

$\mathrm{A}\mathrm{ }=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

$A^2= \left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]= \left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

$A^3=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

$A^4=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Option$\left(1\right)$$A^3+I= A \left(A^3-I\right)$

$\left[\begin{array}{c}0\\ 1\end{array} \begin{array}{c}-1\\ 0\end{array}\right] \left[\begin{array}{c}-1\\ -1\end{array} \begin{array}{c}1\\ -1\end{array}\right]= \left[\begin{array}{c}1\\ -1\end{array} \begin{array}{c}1\\ 1\end{array}\right]$

$\left[\begin{array}{c}1\\ -1\end{array} \begin{array}{c}1\\ 1\end{array}\right]$=$\left[\begin{array}{c}1\\ -1\end{array} \begin{array}{c}1\\ 1\end{array}\right]$.....Correct

Option$\left(2\right)$$A^4-I=A^2+I$

$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]= \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$....Correct

Option$\left(3\right)$$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]= \left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\times \left[\begin{array}{cc}-2& 0\\ 0& -2\end{array}\right] = \left[\begin{array}{cc}0& 2\\ -2& 0\end{array}\right]$.....Incorrect

Option$\left(4\right)$

$A^3-I=A(A-I)$

$\left[\begin{array}{cc}-1& 1\\ -1& -1\end{array}\right]= \left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right] \times \left[\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right] = \left[\begin{array}{cc}-1& 1\\ -1& -1\end{array}\right]$

......Correct.