What is the rank of symmetric Pascal matrix? Can two rows of a symmetric Pascal matrix (in general) be ever linear dependent?
Is the rank of a symmetric Pascal matrix always equal to the number of rows (m)?
I have a symmetric Pascal matrix where m=4 and n=4 and should count its rank.

The symmetric pascal matrix $S$ has $LU$ factorization where $L$ is the lower triangular pascal matrix, and $U$ is the upper triangular matrix.  Both $L$ and $U$ have all ones on the diagonal.  In particular, the determinants of these are each 1.  Thus, using $\det(L)\det(U)=\det(LU)=\det(S)=1$ we can conclude that $S$ is invertible and must have rank=#-columns=#-rows.
In response to your question about dependence of rows:
It is equivalent to consider the columns (since it is symmetric).  By the above reasoning, no two rows/columns of $S$ can be linear combinations of each other.  In fact no $m$ rows or columns (where $1\leq m \leq n$=number of rows = number of columns) can be linearly dependent.
