Let E be an n times n elementary matrix and A be an n times

Let E be an n times n elementary matrix and A be an n times n matrix. Which of the following statements are ALWAYS true? (RS=row space, NS=Null space, CS=column space) NS{EA) = NS(E) RS(EA) = RS(A) CS(EA) = CS{A) RS(EA) = CS{A) NS(A) = NS(AE) (ii) and (iii) (i) and (ii) (ii) and (iv) (i), (ii) and (v) None of the above

Solution

We know Row rank(A)=column rank(A^T), and Columnrank (A)= rowrank(A^T). and Nullty(A)+Rank(A)=n.

So Option D