I've seen 4x4 Matrix used to represent 3D transformations. but last column seems to be always (0,0,0,1).
list of matrices for various operations: Source
Translation and Scaling:
$T = \begin{bmatrix} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ t_{x}& t_{y}& t_{z}& 1\\ \end{bmatrix}$ $S = \begin{bmatrix} S_{x}& 0& 0& 0\\ 0& S_{y}& 0& 0\\ 0& 0& S_{z}& 0\\ 0& 0& 0& 1 \end{bmatrix}$
Rotations:
$R_{x}(\theta) = \begin{bmatrix} 1& 0& 0& 0\\ 0& cos\theta & -sin\theta& 0\\ 0& sin\theta & cos\theta& 0\\ 0& 0& 0& 1\\ \end{bmatrix}$ $R_{y}(\theta) = \begin{bmatrix} cos\theta& 0& sin\theta& 0\\ 0& 1& 0& 0\\ -sin\theta& 0& cos\theta& 0\\ 0& 0& 0& 1\\ \end{bmatrix}$ $R_{z}(\theta) = \begin{bmatrix} cos\theta & -sin\theta & 0& 0\\ sin\theta & cos\theta & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1 \end{bmatrix}$
Shear:
$Sh = \begin{bmatrix} 1& sh_{x}^{y}& sh_{x}^{z}& 0\\ sh_{y}^{x}& 1 & sh_{y}^{z}& 0\\ sh_{z}^{x}& sh_{z}^{y}& 1& 0\\ 0& 0& 0& 1 \end{bmatrix}$
So my question is, can I use 4x3 matrix to reduce calculations? what is the purpose of the last column?
The last column has two purposes. The first is to model vector addition. So normally if I want to find some matrix $A$ so that $Ax = x + a$ for some constant vector $a$ it's not possible. However if we add an extra row and column we can model affine transformations where vector addition can be done with matrix multiplication. This is useful when you're trying to encode local positional information with the translation as you can include it directly into the transform. This is done by replacing the $0$ entries of the last column with the vector $a$ and leaving the $1$ as is.
The second reason is for the projective transform. When calculating vanishing points and other artifacts of binocular vision one step is to homogenize the coordinates, meaning we divide so that one of the coordinate values is $1$. Typically one coordinate is chosen for this so that all points can be compared to each other. For the transform object this will be the $1$ in the bottom right corner. So when you do a projective transform that entry will change and we can homogenize it to get the correct size and orientation of the object.