F={{0,0,0,0},{0,0,1/R,0},{0,-g,0,-g},{0,0,0,0}};
MatrixForm[F]

Part (a)

H={{1,1,0,0}};
MatrixForm[H]
col1=Transpose[H];

col2=Transpose[F].Transpose[H]

col3=Transpose[F].Transpose[F].Transpose[H]

col4=Transpose[F].Transpose[F].Transpose[F].Transpose[H]

Z={{1,0,0,0},{1,0,((-g)/R),0},{0,(1/R),0,((-g)/(R^2))},{0,0,((-g)/R),0}};
MatrixForm[Z]

MatrixRank[Z]

Part (b)

H={{1,1,0,0},{0,0,1,0}};
col1=Transpose[H];
MatrixForm[col1]

col2=Transpose[F].Transpose[H];
MatrixForm[col2]

col3=Transpose[F].Transpose[F].Transpose[H];
MatrixForm[col3]

col4=Transpose[F].Transpose[F].Transpose[F].Transpose[H];
MatrixForm[col4]

ZT={{1,1,0,0},{0,0,1,0},{0,0,(1/R),0},{0,-g,0,-g},{0,((-g)/R),0,((-g)/R)},{0,0,((-g)/R),0},{0,0,((-g)/(R^2)),0},{0,((g^2)/R),0,((g^2)/R)}};
Z=Transpose[ZT];
MatrixForm[Z]

MatrixRank[Z]

Part (c):  We will have the same col1, col2, and col3 as before.  The only thing that changes is the total matrix will not have col4

ZT={{1,1,0,0},{0,0,1,0},{0,0,(1/R),0},{0,-g,0,-g},{0,((-g)/R),0,((-g)/R)},{0,0,((-g)/R),0}};
Z=Transpose[ZT];
MatrixForm[Z]

MatrixRank[Z]