As commentators have mentioned, any matrix produces a linear transformation on vectors through multiplication. In detail using $A$ for your matrix, $[w,x,y,z]A=[(8 x+w (3+x)+3 y+(3+x) z, 8 w+9 x+3 y+7 z, 3 w+3 x+7 y+8 z, 7 x+w (3+x)+8 y+13 z)]$
In fact, the same matrix can represent different transformations depending on the basis being used. If you are certain you are supposed to determine if this is a matrix for a given transformation, then you will need to add information about the basis.
In the case Yrogirg is correct about you wanting it to be a nonsingular transformation, then the course of action would be to compute the determinant and see if/when it is zero. You should get $-54\ell^2+501\ell-1167$. By checking the disriminant you can see that it only has two complex roots, so this matrix is always nonsingular (if you are only interested in real matrices.)
I noticed that the matrix is also symmetric, which I thought might come into the picture somehow.