Question: Find the matrix $\mathbf{M}$ such that
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- \begin{pmatrix} 2 \\ x \\ 3 \end{pmatrix} \cdot \begin{pmatrix} x \\ -1 \\ 4 \end{pmatrix} = 2x + x(-1) + 3(4) = 2x - x + 12 = x + 12 = 0
- Solving:
- x + 12 = 0 \Rightarrow x = \boxed{-12}
- \mathbf{M} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}, \quad \mathbf{M} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}
- Let
- \mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
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