# simple question about matrix

sorry for asking this stupid question. but i cannot come to solution: i am given this matrix and vector.

$$\begin{pmatrix} 1&0&\frac{1}{2}\\ 0&1&0\\ 0&0&0 \end{pmatrix}x=\begin{pmatrix} 0 \\0 \\0 \end{pmatrix}$$

the solution should be $(-1, 0, 2)^T$, but how? this is what i tried:

$$x_1+\frac{x_3}{2}=0 \rightarrow x_1=-\frac{x_3}{2} \\ x_2=0$$ and $x_3=0$. how do i come to the solution of textbook?

thanks a lot

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how is $x_3=0$? – jim Feb 15 '13 at 10:12
@jim, i dont know :(. because $x_3$ is assigned to $0$? – doniyor Feb 15 '13 at 10:13
take $x_3=2\implies x_1 =-1$($x_3$ is arbitrary ) your general solution will be $(\frac{-k}{2},0,k)$ – jim Feb 15 '13 at 10:13

## 1 Answer

Thia equation has infinite no. of solutions.

The equations look like:

$x_1+\frac{x_2}{2}=0\dots (1)$

$x_2=0...(1)$

Apart from these two equations we dont have any other equation (i.e. we dont have any condition on $x_3$) so there is no harm in fixing $x_3$ i.e $x_3=a$ for some $a\in R$

Then all the solutions will be:$(-a/2,0,a)$

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thanks man, i got now – doniyor Feb 15 '13 at 11:02