$$\hat{\beta} = ([inv(X'X)]X')(X\beta + \epsilon)$$
$$\hat{\beta} = \beta + ([inv(X'X)]X')\epsilon$$

$\hat{\beta}$ is an unbiased estimator of $\beta$ under two conditions:

  1. $X$ is non-stochastic $$E(\hat{\beta}) = \beta + E[([inv(X'X)]X')\epsilon]$$ if $X$ is deterministic,  this would reduce to: 
$$E(\hat{\beta}) = \beta + ([inv(X'X)]X') E[\epsilon]$$
The second term on right hand side, $E[\epsilon]$ is zero under one of the Gauss markov assumption. 

  2. $X$ is stochastic but independent of error ($\epsilon$)
Using this, we can reduce the equation to:
$$E(\hat{\beta}) = \beta + inv(X'X)] E[(X')\epsilon]$$
where $E[(X')\epsilon] = 0$ from an assumption that comes from one of the OLS's properties, $E[X'e] = 0$. 

Reference:

https://web.stanford.edu/~mrosenfe/soc_meth_proj3/matrix_OLS_NYU_notes.pdf

Thanks

Anurag