Question

If A = $\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$, then which one of the following statements is not correct ?

• A. $A^3+I=A(A^3-I)$
• B. $A^2+I=A(A^2-I)$
• C. $A^3-I=A(A-I)$
• D. $A^4-I=A^2+I$

Answer 1

Answer: (B)

$A^2+I=A(A^2-I)$

$A=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$
$A^2=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$
$A^3=A^{2}A=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$
$A^4=A^{3}A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
Now, we will check by options.
Option A
$LHS=A^3+I=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 1\\ -1& 1\end{array}\right]$
$RHS=A(A^3-I)=A^4-A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]-\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}1& 1\\ -1& 1\end{array}\right]$
Here, $LHS=RHS$
Hence, statement given in option A is correct.
Option B,
$LHS=A^2+I=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$RHS=A(A^2-I)=A^3-A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]-\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}0& 2\\ -2& 0\end{array}\right]$
Here, $LHS\ne RHS$
Hence, statement given in option B is incorrect.
Option C,
$LHS=A^3-I=\left[\begin{array}{cc}-1& 1\\ -1& -1\end{array}\right]$
$RHS=A(A-I)=\left[\begin{array}{cc}-1& 1\\ -1& -1\end{array}\right]$
$LHS=RHS$
Option D
$LHS=(A^4-I)=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$RHS=(A^2+I)=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$LHS=RHS$

Hence, only option $B$ is incorrect.