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)

$\left[ \begin{matrix} -3 & 0 \\\ 0 & -3 \\\ \end{matrix} \right]$

Answer 2

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
$X=-3I=\left[ \begin{matrix} -3 & 0 \\\ 0 & -3 \\\ \end{matrix} \right]$