Question
If \[ [1\ \ 1\ \ x] \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = 0, \] find \(x\).
Solution (Single Matrix Equation Method)
\[ [1\ \ 1\ \ x] \begin{bmatrix} 1+0+2 \\ 0+2+1 \\ 2+1+0 \end{bmatrix} = [1\ \ 1\ \ x] \begin{bmatrix} 3 \\ 3 \\ 3 \end{bmatrix} = 1\cdot3 + 1\cdot3 + x\cdot3 = 0 \] \[ 3 + 3 + 3x = 0 \] \[ 6 + 3x = 0 \] \[ x = -2 \] —Final Answer
\[
x = -2
\]
Hence, the required value is \(x = -2\).