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.Please answer the two red boxes.Chapter 4.1 Question 2 A square matrix A is nilpotent if A"= 0 for some positive integer n.Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 x 2 nilpotent matrices with real entries. Is H asubspace of the vector space V?1. Is H nonempty?H is nonempty2. 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[1 2] [56separated list and syntax such as [[1,2],[3,4]], [[5,6],[7,8]] for the answerl: (Hint: to show that H is not closed7under addition, it is sufficient to find two nilpotent matrices A and B such that (A + B)" + 0 for all positive integers n.)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[3in H, using a comma separated list and syntax such as 2, [[3,4] , [5,6]] for the answer 2,4(Hint: to show that H is notclosed under scalar multiplication, it is sufficient to find a real number r and a nilpotent matrix A such that (rA)" + 0 for all positiveintegers n.)

Answered: Image /qna-images/answer/39f39786-25d2-45a9-870e-dc23f13459b3.jpg

A square matrix A is nilpotent if A" = 0 for some positive integer n. Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 × 2 nilpotent matrices with real entries. Is H a subspace of the vector space V? 1. Is H nonempty? H is nonempty ♥ O 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 answer 6] (Hint: to show that H is not closed under addition, it is sufficient to find two nilpotent matrices A and B such that (A + B)" + 0 for all positive integers n.) 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, (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and a nilpotent matrix A such that (rA)" + 0 for all positive integers n.)

A square matrix A is nilpotent if A" = 0 for some positive integer n. Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 × 2 nilpotent matrices with real entries. Is H a subspace of the vector space V? 1. Is H nonempty? H is nonempty ♥ O 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 answer 6] (Hint: to show that H is not closed under addition, it is sufficient to find two nilpotent matrices A and B such that (A + B)" + 0 for all positive integers n.) 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, (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and a nilpotent matrix A such that (rA)" + 0 for all positive integers n.)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.2: Matrix Algebra

Problem 29EQ: A square matrix is called upper triangular if all of the entries below the main diagonal are zero....

See similar textbooks