Question
Solve the matrix equation: \[ [2x\ \ 3] \begin{bmatrix} 1 & 2 \\ -3 & 0 \end{bmatrix} \begin{bmatrix} x \\ 8 \end{bmatrix} = 0 \]
Solution
Step 1: Single Matrix Equation
\[ [2x\ \ 3] \begin{bmatrix} x + 16 \\ -3x \end{bmatrix} = 2x(x+16) + 3(-3x) = 0 \]Step 2: Simplify
\[ 2x^2 + 32x – 9x = 0 \] \[ 2x^2 + 23x = 0 \]Step 3: Solve
\[ x(2x + 23) = 0 \] \[ x = 0 \quad \text{or} \quad x = -\frac{23}{2} \]Final Answer
\[
x = 0 \quad \text{or} \quad x = -\frac{23}{2}
\]