For what values of `x : [[1, 2, 1]][[1 ,2 ,0],[ 2, 0, 1],[ 1, 0, 2]][[0],[ 2],[x]]=O`?

Explanation

To solve the equation \(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}\), we will follow these steps:

### Step 1: Matrix Multiplication
First, we will multiply the row matrix \(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\) with the square matrix \(\begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix}\).

\[ \begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} = \begin{bmatrix} 1 \cdot 1 + 2 \cdot 2 + 1 \cdot 1 & 1 \cdot 2 + 2 \cdot 0 + 1 \cdot 0 & 1 \cdot 0 + 2 \cdot 1 + 1 \cdot 2 \end{bmatrix} \]
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