Find x_i if [ lll 1 & x & 1 ] [ lll 1 & 2 & 3 4 & 5 & 6 3 & 2 & 5… - Vidyadip

Question

Find $x_i$ if $\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5\end{array}\right]\left[\begin{array}{c}1 \\ -2 \\ 3\end{array}\right]=0$

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Answer

$\begin{aligned} & {\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5\end{array}\right]\left[\begin{array}{c}1 \\ -2 \\ 3\end{array}\right]=0} \\ & \therefore \quad\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{c}1-4+9 \\ 4-10+18 \\ 3-4+15\end{array}\right]=0 \\ & \therefore \quad\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{c}6 \\ 12 \\ 14\end{array}\right]=0 \\ & \end{aligned}$

∴ [6 + 12x + 14] =[0] ∴ By equality of matrices, we get ∴ 12x + 20 = 0 ∴ 12x =-20

$\therefore x=\frac{-5}{3}$

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