Question

Given $A=\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right];B=\left[\begin{array}{cc}9& 3\\ 3& 1\end{array}\right]$. $I$ is a unit matrix of order 2. Find all possible matrix $X$ if $AX=O$ but $BX\ne O$

• A. $X$ does not exist
• B. $X$ is a unit matrix
• C. $X$ is a identity matrix
• D. None of these

Answer 1

The correct option is A $X$ does not exist

$A$ = $\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$ and $B$ = $\left[\begin{array}{cc}9& 3\\ 3& 1\end{array}\right]$
let $X$ = $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
given $AX=O$ and $BX\ne O$
$\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\Rightarrow 2a+c=0;2b+d=0;a+c=0;b+d=0\Rightarrow a=b=c=d=0$
i.e $X=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$\Rightarrow BX=0$ but $BX\ne 0$
Hence, $X$ does not exist.