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Find the value of $x$ if $\begin{bmatrix}1 & x &1\end{bmatrix}\begin{bmatrix}1 & 3 & 2\\2 & 5 & 1\\15 & 3 & 2\end{bmatrix}\begin{bmatrix}1\\2\\x\end{bmatrix}=0$.

Toolbox:
• If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
Step1:
Consider $\begin{bmatrix}1 & 3 & 2\\2 & 5 & 1\\15 & 3 & 2\end{bmatrix}\begin{bmatrix}1\\2\\x\end{bmatrix}$
$\;\;\;=\begin{bmatrix}1+6+2x\\2+10+x\\15+6+2x\end{bmatrix}$ (we get this by multiplying the column with each row .)
$\;\;\;=\begin{bmatrix}2x+7\\12+x\\21+2x\end{bmatrix}$
Step2:
Now consider
$\begin{bmatrix}1 & x & 1\end{bmatrix}\begin{bmatrix}2x+7\\12+x\\21+2x\end{bmatrix}$
1(2x+7)+x(12+x)+1(21+2x)=0.
$2x+7+12x+x^2+21+2x=0.$
$x^2+16x+28=0.$
$x^2+14x+2x+28$=0
x(x+14)+2(x+14)=0.
(x+2)(x+14)=0.
x=-2,-14.
Value of x=-2,-14.