Question
If \[ [x\ \ 4\ \ 1] \begin{bmatrix} 2 & 1 & 2 \\ 1 & 0 & 2 \\ 0 & 2 & -4 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ -1 \end{bmatrix} = 0, \] find \(x\).
Solution
Step 1: Single Matrix Equation
\[ [x\ \ 4\ \ 1] \begin{bmatrix} 2x + 4 – 2 \\ x – 2 \\ 12 \end{bmatrix} = x(2x+2) + 4(x-2) + 12 = 0 \]Step 2: Simplify
\[ 2x^2 + 2x + 4x – 8 + 12 = 0 \] \[ 2x^2 + 6x + 4 = 0 \]Step 3: Solve
\[ x^2 + 3x + 2 = 0 \] \[ (x+1)(x+2)=0 \]Final Answer
\[
x = -1 \quad \text{or} \quad x = -2
\]