# Find the value of x, if : $\begin{bmatrix} 1 & x & 1 \end{bmatrix} \; \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5 \end{bmatrix} \; \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix} = 0$

## 1 Answer

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:
Given:
$\begin{bmatrix} 1 & x & 1 \end{bmatrix} \left \{\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5 \end{bmatrix} \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix}\right\} = 0$
$\begin{bmatrix}1 & x &1\end{bmatrix}\begin{bmatrix}1(1)+2(-2)+3(3)\\4(1)+5(-2)+6(3)\\3(1)+2(-2)+5(3)\end{bmatrix}=0$
$\begin{bmatrix}1 & x &1\end{bmatrix}\begin{bmatrix}1-4+9\\4-10+18\\3-4+15\end{bmatrix}=0$
$\begin{bmatrix}1 & x &1\end{bmatrix}\begin{bmatrix}6\\12\\14\end{bmatrix}=0$
Step2:
$1(6)+x(12)+1(14)=0$
$6+12x+14=0$
$12x=20$
$x=\Large \frac{20}{12}$
$x=\Large \frac{4}{3}$
answered Apr 9, 2013

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