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Question
If [x 4 1] `[[2 1 2],[1 0 2],[0 2 -4]]` `[[x],[4],[-1]]` = 0, find x.
Solution
\[Given: \begin{bmatrix}x & 4 & 1\end{bmatrix}\begin{bmatrix}2 & 1 & 2 \\ 1 & 0 & 2 \\ 0 & 2 & - 4\end{bmatrix}\begin{bmatrix}x \\ 4 \\ - 1\end{bmatrix} = 0\]
\[ \Rightarrow \begin{bmatrix}2x + 4 + 0 & x + 0 + 2 & 2x + 8 - 4\end{bmatrix}\begin{bmatrix}x \\ 4 \\ - 1\end{bmatrix} = 0\]
\[ \Rightarrow \begin{bmatrix}2x + 4 & x + 2 & 2x + 4\end{bmatrix}\begin{bmatrix}x \\ 4 \\ - 1\end{bmatrix} = 0\]
\[ \Rightarrow \begin{bmatrix}2 x^2 + 4x + 4x + 8 - 2x - 4\end{bmatrix} = 0\]
\[ \Rightarrow 2 x^2 + 6x + 4 = 0\]
\[ \Rightarrow x^2 + 3x + 2 = 0\]
\[ \Rightarrow x^2 + 2x + x + 2 = 0\]
\[ \Rightarrow x\left( x + 2 \right) + 1\left( x + 2 \right) = 0\]
\[ \Rightarrow \left( x + 2 \right)\left( x + 1 \right) = 0\]
\[ \Rightarrow x + 2 =\text{ 0 or x + 1}= 0\]
\[ \Rightarrow x =\text{ - 2 or x} = - 1\]
\[ \therefore x = - 2, - 1\]
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