Question
Solve the matrix equation: \[ [1\ \ 2\ \ 1] \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix} = 0 \]
Solution
Step 1: Single Matrix Equation
\[ [1\ \ 2\ \ 1] \begin{bmatrix} 4 \\ x \\ 2x \end{bmatrix} = 1\cdot4 + 2\cdot x + 1\cdot 2x = 0 \]Step 2: Simplify
\[ 4 + 2x + 2x = 0 \] \[ 4 + 4x = 0 \]Step 3: Solve
\[ 4x = -4 \] \[ x = -1 \]Final Answer
\[
x = -1
\]