Question

If $A=\left[\begin{array}{ccc}1& -1& 1\\ 2& -1& 0\\ 1& 0& 0\end{array}\right]$, then

• A. $A^3=I$
• B. $A^{-1}=A^2$
• C. $A^n=A\forall n\ne 4$
• D. none of these

Answer 1

Answer: (A), (B)

$A^3=I$
$A^{-1}=A^2$

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

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

$A^3=\left[\begin{array}{ccc}0& 0& 1\\ 0& -1& 2\\ 1& -1& 1\end{array}\right]\left[\begin{array}{ccc}1& -1& 1\\ 2& -1& 0\\ 1& 0& 0\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$\Rightarrow A^3=I$

$\Rightarrow A^2=A^{-1}$ and $A^5=A^2\ne A$

Hence, options A and B.