Question
If \[ [1\ \ -1\ \ x] \begin{bmatrix} 0 & 1 & -1 \\ 2 & 1 & 3 \\ 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = 0, \] find \(x\).
Solution
Step 1: Single Matrix Equation
\[ [1\ \ -1\ \ x] \begin{bmatrix} 0 + 1 – 1 \\ 0 + 1 + 3 \\ 0 + 1 + 1 \end{bmatrix} = [1\ \ -1\ \ x] \begin{bmatrix} 0 \\ 4 \\ 2 \end{bmatrix} = 1\cdot0 + (-1)\cdot4 + x\cdot2 = 0 \]Step 2: Simplify
\[ -4 + 2x = 0 \]Step 3: Solve
\[ 2x = 4 \] \[ x = 2 \]Final Answer
\[
x = 2
\]