If $\begin{bmatrix} 1 & 2 & 1\end{bmatrix}$ $\begin{bmatrix}1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x\end{bmatrix} = 0,$then value of x is

The given matrix equation is:

$\begin{bmatrix} 1 & 2 & 1\end{bmatrix} \begin{bmatrix}1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x\end{bmatrix} = 0$

Multiply the first two matrices from left to right.

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$\begin{bmatrix} 1 & 2 & 1\end{bmatrix} \times \begin{bmatrix}1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix}$

First element: $(1 \times 1) + (2 \times 2) + (1 \times 1)$

$= 1 + 4 + 1$

$= 6$

Second element: $(1 \times 2) + (2 \times 0) + (1 \times 0)$

$= 2 + 0 + 0$

$= 2$

Third element: $(1 \times 0) + (2 \times 1) + (1 \times 2)$

$= 0 + 2 + 2$

$= 4$

Result: $\begin{bmatrix} 6 & 2 & 4\end{bmatrix}$

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$\begin{bmatrix} 6 & 2 & 4\end{bmatrix} \times \begin{bmatrix} 0 \\ 2 \\ x\end{bmatrix} = 0$

$(6 \times 0) + (2 \times 2) + (4 \times x) = 0$

$0 + 4 + 4x = 0$

$4 + 4x = 0$

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$4x = -4$

$x = \dfrac{-4}{4}$

$x = -1$

Therefore, the value of $x = -1$.