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$x,y$ અને $z$ ની કિમત મેળવો : $\left[\begin{array}{ll}x+y & 2 \\ 5+z & x y\end{array}\right]=\left[\begin{array}{ll}6 & 2 \\ 5 & 8\end{array}\right]$
$x=2$, $y=2,$ $z=0$ or $x=4$, $y=4,$ $z=0$
$x=4$, $y=4,$ $z=0$ or $x=2$, $y=2,$ $z=0$
$x=0$, $y=2,$ $z=0$ or $x=2$, $y=0,$ $z=0$
$x=4$, $y=2,$ $z=0$ or $x=2$, $y=4,$ $z=0$
Solution
$\left[\begin{array}{ll}x+y & 2 \\ 5+z & x y\end{array}\right]=\left[\begin{array}{ll}6 & 2 \\ 5 & 8\end{array}\right]$
As the given matrices are equal, their corresponding elements are also equal. Comparing the corresponding elements, we get:
$x+y=6$, $x y=8$, $5+z=5$
Now, $5+z=5 \Rightarrow z=0$
Using $(x-y)^{2}=(x+y)^{2}-4 x y,$ we get
$\Rightarrow(x-y)^{2}=36-32=4$
$\Rightarrow $ $x-y=\pm 2$
When $x-y=2$ and $x+y=6,$ we get $x=4$ and $y=2$
When $x-y=-2$ and $x+y=6$ we get $x=2$ and $y=4$
$\therefore $ $x=4$, $y=2,$ and $z=0$ or $x=2$, $y=4,$ and $z=0$
