Question

Find x,y and z, so that A=B, where
$A=\begin{bmatrix}x-2&3&2z\\ 18z&y+2&6z\end{bmatrix} ,\ B=\begin{bmatrix}y&z&6\\ 6y&x&2y\end{bmatrix}$.

Answer 1

Hint: In this question, we have to find the value of x,y and z. So firstly we have to use the given condition i.e, A=B, since the dimensions or orders of the given matrices A and B are equal (i.e $2\times 3$ ) so we can find 6 linear equations by equating the corresponding elements of the given matrices.

Complete step-by-step answer:
So our given condition is
A=B………………………………. equation(1)
Now putting the values of Matrix A and B in equation(1), we get,
$\begin{bmatrix}x-2&3&2z\\ 18z&y+2&6z\end{bmatrix} =\begin{bmatrix}y&z&6\\ 6y&x&2y\end{bmatrix}$
Now equating the corresponding elements, we get,
x-2 = y ………... equation(2)
3 = z …………... equation(3)
2z = 6 …………..equation(4)
18z = 6y ………..equation(5)
y+2 = x ………….equation(6)
6z = 2y ………….equation(7)

Now from equation(3), we can easily say that the value of z is 3,
i.e z = 3.
Now putting the value of z in equation(7), we get,
$6\times 3=2y$
$\Rightarrow 18=2y$
$\Rightarrow 2y=18$
On dividing 2 on both the side,
$\Rightarrow y=\dfrac{18}{2}$
$\Rightarrow y=9$
So from the above we get the value of y is 9,
i.e y=9.
Now by putting the value of y in equation(1), we get,
$\Rightarrow x-2=9$
$\Rightarrow x=9+2$
$\Rightarrow x=11$
So ultimately we get the value of x, which is 11.

So our required solutions are x=11, y=9, z=3.

Note: In this type of question, you should know that if you have given two matrices which are equal and if you are being asked to find the values of elements then you have to solve by equating the corresponding elements of those given matrices.