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If $X$ and $Y$ are the matrices of order $2 \times 2$ each and
$2 X-3 Y=\left[\begin{array}{cc}-7 & 0 \\ 7 & -13\end{array}\right]$ and $3 X+2 Y=\left[\begin{array}{ll}9 & 13 \\ 4 & 13\end{array}\right]$, then what is $Y$ equal to?
Options:
$2 X-3 Y=\left[\begin{array}{cc}-7 & 0 \\ 7 & -13\end{array}\right]$ and $3 X+2 Y=\left[\begin{array}{ll}9 & 13 \\ 4 & 13\end{array}\right]$, then what is $Y$ equal to?
Solution:
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Verified Answer
The correct answer is:
$\left[\begin{array}{cc}3 & 2 \\ -1 & 5\end{array}\right]$
Let $X$ and $Y$ be two matrices of order $2 \times 2$ each.
Given, $2 X-3 Y=\left[\begin{array}{cc}-7 & 0 \\ 7 & -13\end{array}\right]$ ...(i)
and $3 X+2 Y=\left[\begin{array}{ll}9 & 13 \\ 4 & 13\end{array}\right]$ ...(ii)
On multiplying Eq. (i) by 3 and Eq. (ii) by 2, we get
$6 X-9 Y=\left[\begin{array}{cc}-21 & 0 \\ 21 & -39\end{array}\right]$ ...(iii)
$6 X+4 Y=\left[\begin{array}{cc}18 & 26 \\ 8 & 26\end{array}\right]$ ...(iv)
On subtracting Eqs. (iii) from (iv), we get
$13 Y=\left[\begin{array}{cc}39 & 26 \\ -13 & 65\end{array}\right] \Rightarrow Y=\left[\begin{array}{cc}3 & 2 \\ -1 & 5\end{array}\right]$
Given, $2 X-3 Y=\left[\begin{array}{cc}-7 & 0 \\ 7 & -13\end{array}\right]$ ...(i)
and $3 X+2 Y=\left[\begin{array}{ll}9 & 13 \\ 4 & 13\end{array}\right]$ ...(ii)
On multiplying Eq. (i) by 3 and Eq. (ii) by 2, we get
$6 X-9 Y=\left[\begin{array}{cc}-21 & 0 \\ 21 & -39\end{array}\right]$ ...(iii)
$6 X+4 Y=\left[\begin{array}{cc}18 & 26 \\ 8 & 26\end{array}\right]$ ...(iv)
On subtracting Eqs. (iii) from (iv), we get
$13 Y=\left[\begin{array}{cc}39 & 26 \\ -13 & 65\end{array}\right] \Rightarrow Y=\left[\begin{array}{cc}3 & 2 \\ -1 & 5\end{array}\right]$
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