Given: 3 [ cc x & y z & w ]= [ cc x & 6 -1 & 2 w ]+ [ cc 4 & x+y … - Vidyadip

Question

Given: $3\left[\begin{array}{cc}x & y \\ z & w\end{array}\right]=\left[\begin{array}{cc}x & 6 \\ -1 & 2 w\end{array}\right]+\left[\begin{array}{cc}4 & x+y \\ z+w & 3\end{array}\right]$, find the values of $\mathrm{x}, \mathrm{y}, \mathrm{z}$ and $\mathrm{w}$.

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Answer

Given: $3\left[ {\begin{array}{*{20}{c}} x&y \\ z&w \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} x&6 \\ { - 1}&{2w} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 4&{x + y} \\ {z + w}&3 \end{array}} \right]$
$\Rightarrow \left[ {\begin{array}{*{20}{c}} {3x}&{3y} \\ {3z}&{3w} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {x + 4}&{6 + x + y} \\ { - 1 + z + w}&{2w + 3} \end{array}} \right]$
Equating corresponding entries, we have
3x = x + 4 $\Rightarrow$ 2x = 4 $\Rightarrow$ x = 2
and 3y = 6 + x + y
$\Rightarrow$ 2y = 6 + 2
$\Rightarrow$ 2y = 8
$\Rightarrow$ y = 4
and 3z = -1 + z + w $\Rightarrow$ 2z - w = - 1 ….(i)
and 3w = 2w + 3 $\Rightarrow$ w = 3
Putting w = 3 in eq. (i), 2z - 3 = -1
$\Rightarrow$ 2z = 2 $\Rightarrow$ z = 1
$\therefore$ x = 2, y = 4, z = 1, w = 3

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