Question

If $X$ and $Y$ are $ 2\times 2 $ matrices such that $ 2X+3Y=O $ and $ X+2Y=I, $ where $ O $ and $ I $ denote the $ 2\times 2 $ zero matrix and the $ 2\times 2 $ identity matrix, then $X$ is equal to

• A. $ \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] $
• B. $ \left[ \begin{matrix} 2 & 0 \\ 0 & 2 \\ \end{matrix} \right] $
• C. $ \left[ \begin{matrix} -3 & 0 \\ 0 & -3 \\ \end{matrix} \right] $
• D. $ \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \\ \end{matrix} \right] $

Answer 1

Answer: (C)

Given, $ 2X+3Y=O $ .....(i) and $ X+2Y=I $ ..(ii)
Where $ O=\left[ \begin{matrix} 0 & 0 \\ 0 & 0 \\ \end{matrix} \right] $ and $ I=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] $
On solving Eqs. (i) and (ii), we get
$ X=-3I=\left[ \begin{matrix} -3 & 0 \\ 0 & -3 \\ \end{matrix} \right] $