Question

$A=\left(\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 0& 1& 0\end{array}\right)\Rightarrow A^2-2A=$

• A. $A^{-1}$
• B. $-A^{-1}$
• C. $I$
• D. $-I$

Answer 1

The correct option is B $-A^{-1}$

Given ,
$A=\left(\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 0& 1& 0\end{array}\right)$
To find
$A^2-2A$ =
We have ,
$A^2=A.A$
$=\left(\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 0& 1& 0\end{array}\right).\left(\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 0& 1& 0\end{array}\right)$

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

$A^2-2A=\left(\begin{array}{ccc}-1& 1& -1\\ 0& 0& -1\\ 0& -1& 1\end{array}\right)$
And this is equal to $A^{-1}$

$A^{-1}=-\left(\begin{array}{ccc}-1& 1& -1\\ 0& 0& -1\\ 0& -1& 1\end{array}\right)$
Therefore, $A^2-2A=-A^{-1}$
Hence, option B