A square matrix A is idempotent if A^2 = A.

Let V be the vector space of all 2 \times 2 matrices with real entries. Let H be the set of all 2 \times 2 idempotent matrices with real entries. Is H a subspace of the vector space V?

1. Does H contain the zero vector of V?

2. choose

3. H contains the zero vector of V

4. H does not contain the zero vector of V




5. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is not in H, using a comma separated list and syntax such as \verb+[[1,2],[3,4]], [[5,6],[7,8]]+ for the answer \left[\begin{array}{cc} 1 \amp 2\cr 3 \amp 4 \end{array}\right],\left[\begin{array}{cc} 5 \amp 6\cr 7 \amp 8 \end{array}\right]. (Hint: to show that H is not closed under addition, it is sufficient to find two idempotent matrices A and B such that (A+B)^2 \ne (A+B).)

   
   
   

6. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in \mathbb{R} and a matrix in H whose product is not in H, using a comma separated list and syntax such as \verb+2, [[3,4],[5,6]]+ for the answer 2, \left[\begin{array}{cc} 3 \amp 4\cr 5 \amp 6 \end{array}\right]. (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)^2 \ne (rA).)

   
   
   

7. Is H a subspace of the vector space V? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3.

8. choose

9. H is a subspace of V

10. H is not a subspace of V