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# For what values of $x$, $\begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix}$ = 0 ?

$\begin{array}{1 1} x=2 \\ x=1 \\ x=-1 \\ x=3 \end{array}$

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: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
Step 1: LHS
$\begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix}$
$\Rightarrow \begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 0+4+0 \\ 0+0+x \\ 0+0+2x\end{bmatrix}$
$\Rightarrow \begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 4 \\ x \\ 2x\end{bmatrix}$
$\Rightarrow [4+2x+2x]\Rightarrow [4+4x]$
Step 2: Now equate the obtained LHS value to the given RHS
[4+4x]=[0]
$\Rightarrow$ 4+4x=0
4x=-4
x=-1.
answered Mar 4, 2013
edited Mar 19, 2013