Question
Solve the matrix equation: \[ [x\ \ 1] \begin{bmatrix} 1 & 0 \\ -2 & -3 \end{bmatrix} \begin{bmatrix} x \\ 5 \end{bmatrix} = 0 \]
Solution
Step 1: Single Matrix Equation
\[ [x\ \ 1] \begin{bmatrix} x \\ -2x – 15 \end{bmatrix} = x(x) + 1(-2x – 15) = 0 \]Step 2: Simplify
\[ x^2 – 2x – 15 = 0 \]Step 3: Solve
\[ (x – 5)(x + 3) = 0 \] \[ x = 5 \quad \text{or} \quad x = -3 \]Final Answer
\[
x = 5 \quad \text{or} \quad x = -3
\]