Find x, y and z so that A = B, where
Question:

Find $x, y$ and $z$ so that $A=B$, where

$A=\left[\begin{array}{ccc}x-2 & 3 & 2 z \\ 18 z & y+2 & 6 z\end{array}\right], B=\left[\begin{array}{ccc}y & z & 6 \\ 6 y & x & 2 y\end{array}\right]$

Solution:

Since all the corresponding elements of a matrix are equal,

$A=\left[\begin{array}{ccc}x-2 & 3 & 2 z \\ 18 z & y+2 & 6 z\end{array}\right], B=\left[\begin{array}{ccc}y & z & 6 \\ 6 y & x & 2 y\end{array}\right]$

Here,

$x-2=y \quad \ldots(1)$

$z=3 \quad \ldots(2)$

$18 z=6 y \quad \ldots(3)$

Putting the value of $z$ in eq. (3), we get

$18(3)=6 y$

$\Rightarrow 54=6 y$

$\Rightarrow y=\frac{54}{6}=9$

Putting the value of $y$ in eq. (1), we get

$x-2=9$

$\Rightarrow x=9+2$

$\Rightarrow x=11$

$\therefore x=11, y=9$ and $z=3$