HW 15 Sec 4.3: Problem 21 Previous Problem List Next (1 point) Note: This proble

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HW 15 Sec 4.3: Problem 21 Previous Problem List Next (1 point) Note: This problem is extra credit! Correct answers will contribute to your WeBWorK average, but you are not required to do this problem for full homework credit. Homework averages over 100% ARE allowed. A square matrix A is nilpotent if A" for some positive integer n. Let V be the vector space of all 2 vector space V? 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 1. Is H nonempty? choose 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 1 2 [5 6 syntax such as 1.211341l115.6,17.8l) for the answer3 find two nilpotent matrices A and B such that (A + B)"0 for all positive integers n.) Hint: to show that H is not closed under addition, it is sufficient to 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 nd syntax such as 2.113,4115,6] orth aswer 2 6 multiplication, it is sufficient to find a real number r and a nilpotent matrix A such that (rA Hint: to show that H is not closed under scalar · 0 for all positive ntegersn.) 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. choose Note: In order to get creait for this problem al answers must be correct. Preview My Answers Submit Answers

Explanation / Answer

2

-2

2

-2

            and B =

0

1

0

0

             belong to H as A2 = 0 and B2= 0.

             Further, the 2x2 zero matrix is also in H.

2. Let H be the set of all 2x2 nilpotent matrices.Let A and B be two arbitrary elements of H. Since , (A+B)p ( p is a positive integer) need not be equal to Ap+Bp, in general, hence A+B, need not bea nilpotent matrix. Thus, H is not closed under vector addition. The matrices A and B in par 1 above are nilpotent matrices, but A+B =

2

-1

2

-2

              is not a nilpotent matrix ( as det(A+B) 0).

3. H is closed under scalar multiplication. If A is a 2x2 nilpotent matrix such that An= 0 and if is a real valued scalar, then (A)n = nAn = 0.

4. H is not a vector space as it is not closed under vector addition.

2

-2

2

-2

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