The rows of a matrix are linerarly dependent if and only if the determinant of the matrix is zero. (The same goes for the columns).

You probably know that if k is a number and A is a matrix, then k*det(A) is equal the the determinant of the matrix you get if you multiply one of the rows of A with k. Also, det(A) is the same as the determinant of the matrix you get if you take one row of A and add it to another. When the rows are linearly dependent you can use these facts to see that det(A)=(some number)*(the determinant of a matrix with all zeros on one of the rows)=0.