can anyone tell me if at all there is any way in which rank of a matrix can be equal to 0?? and if P(A)!=P(adjA)..???
The rank of a matrix (or, more generally, linear transformation) is the dimension of its image. As long as the matrix A is non-zero (has any non-zero components), it will map some non-zero vector to a non-zero vector and any multiple of that is in its image so the image has dimension at least 1. The only matrix with rank 0 is the zero matrix. Whether the matrix is symmetric or not is irrelevant.