Please answer correctly following the same format as the questions. Box the correct answer and I will give a thumbs up if the answer is correct, a thumbs down if answer is incorrect.Chapter 4.1 Question 6Please answer the two red boxes You have answered 2 out of 4 parts correctly.A square matrix A is idempotent if A² = A.Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 × 2 idempotentmatrices 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 V2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum isnot in H, using a comma separated list and syntax such as [[1,2],[3,4]], [[5,6], [7,8]] for the2]answer[158]. (Hint: to show that H is not closed under addition, it is sufficient to find two3idempotent matrices A and B such that (A + B)² ‡ (A + B).)closed3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and amatrix in H whose product is not in H, using a comma separated list and syntax such as2, [[3,4], [5,6]] for the answer 2, [34]. (Hint: to show that H is not closed under scalar5multiplication, it is sufficient to find a real number r and an idempotent matrix A such that(rA)² + (rA).)closed4. Is H a subspace of the vector space V? You should be able to justify your answer by writing acomplete, coherent, and detailed proof based on your answers to parts 1-3.H is not a subspace of V

As a matrix A is said to be idempotent if A2 = A

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] answer [33] [58]- (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 ‡ (A + B).) closed 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 2, [[3,4], [5,6]] for the answer 2, [36]. (Hint: to show that H is not closed under scalar 5 multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)² + (rA).) closed 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

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] answer [33] [58]- (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 ‡ (A + B).) closed 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 2, [[3,4], [5,6]] for the answer 2, [36]. (Hint: to show that H is not closed under scalar 5 multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)² + (rA).) closed 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

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.6: Rank Of A Matrix And Systems Of Linear Equations

Problem 67E: Let A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many...

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