Question
Solve the matrix equation: \[ [x\ \ -5\ \ -1] \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ 1 \end{bmatrix} = 0 \]
Solution
Step 1: Single Matrix Equation
\[ [x\ \ -5\ \ -1] \begin{bmatrix} x + 2 \\ 8 + 1 \\ 2x + 3 \end{bmatrix} = [x\ \ -5\ \ -1] \begin{bmatrix} x + 2 \\ 9 \\ 2x + 3 \end{bmatrix} \] \[ = x(x+2) – 5(9) – 1(2x+3) = 0 \]Step 2: Simplify
\[ x^2 + 2x – 45 – 2x – 3 = 0 \] \[ x^2 – 48 = 0 \]Step 3: Solve
\[ x^2 = 48 \] \[ x = \pm 4\sqrt{3} \]Final Answer
\[
x = 4\sqrt{3} \quad \text{or} \quad x = -4\sqrt{3}
\]