When we translate a point $p_3 = (x,y,z)$ to coordinates $p_4 =(x + t_x , y + t_y ,z + t_z,1)$ we use $4 \times 4$ Translation matrix using homogenous coordinates, hence we add a $1$ to fourth coordinate of $p_4$.
But I could not find a matrix $M : R^3 \rightarrow R^4$ for which $M p_3 = p_4$. Note that I am talking about a $1$ on the fourt coordinate what ever vector $p_3$ be.
Why can not we find such a generic translation matrix. Omit the case of matrix addition i.e. $M$ may only be multiplication of other matrices.