Question

which of 2×2 matrix is linearly independent ​

Answer 1

Linear independence of matrices is essentially their linear independence as vectors. So you are trying to show that the vectors (1,−1,0,2),(0,1,3,0),(1,0,1,0) and (1,1,1,1) are linearly independent.

These are precisely the rows of the matrix that you have given. If this matrix is indeed row equivalent to the identity matrix (a fact which I'm assuming) then the vector space the above four vectors will generate will have dimension four (recall that, row or column operations don't change the rank of a matrix). This shows that they are linearly independent.