Please solve and answer in the same format as the question as it makes it easier to follow. Box answers and do not type it out and I will give you a thumbs up rating if answered correctly.
Chapter 4.1 Question 6
**Part two and three are answered incorrectly. Please tell me what to put in the boxes.**
![A square matrix A is idempotent if A? = A.<br>Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 idempotent<br>matrices with real entries. Is H a subspace of the vector space V?<br>1. Does H contain the zero vector of V?<br>H contains the zero vector of V<br>2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is<br>not in H, using a comma separated list and syntax such as [[1,2], [3,4]], [[5,6],[7,8]] for the<br>[1 2] [5 6]<br>3<br>E 1E :<br>(Hint: to show that H is not closed under addition, it is sufficient to find two<br>answer<br>7 8<br>idempotent matrices A and B such that (A + B)? + (A + B).)<br>[1.0].[0,0].[1.0].[0,1]<br>3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a<br>matrix in H whose product is not in H, using a comma separated list and syntax such as<br>[3 4]<br>2, [[3,4], [5,6]] for the answer 2,<br>:1. (Hint: to show that H is not closed under scalar<br>5 6<br>multiplication, it is sufficient to find a real number r and an idempotent matrix A such that<br>(rA)² + (rA).)<br>2([1,0].[0,0])<br>4. Is H a subspace of the vector space V? You should be able to justify your answer by writing a<br>complete, coherent, and detailed proof based on your answers to parts 1-3.<br>H is not a subspace of V ♥ O<br>](https://s3.us-east-1.amazonaws.com/storage.unifolks.com/qimg-008/008_e34cx50-on5cc2oq.png)
Extracted text: A square matrix A is idempotent if A? = A. Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 idempotent matrices with real entries. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? H contains the zero vector of V 2. 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 [[1,2], [3,4]], [[5,6],[7,8]] for the [1 2] [5 6] 3 E 1E : (Hint: to show that H is not closed under addition, it is sufficient to find two answer 7 8 idempotent matrices A and B such that (A + B)? + (A + B).) [1.0].[0,0].[1.0].[0,1] 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a matrix in H whose product is not in H, using a comma separated list and syntax such as [3 4] 2, [[3,4], [5,6]] for the answer 2, :1. (Hint: to show that H is not closed under scalar 5 6 multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)² + (rA).) 2([1,0].[0,0]) 4. 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. H is not a subspace of V ♥ O