Let A be a skew-symmetric matrix that A T = -A,assume that A is an nxn matrix 1)

Question

Let A be a skew-symmetric matrix that AT= -A,assume that A is an nxn matrix 1) show that I+A is invertible 2)show that P=(I-A)(I+A)-1 is orthogonal 3) show that every orthogonal matrix P such that I+P isinvertible arises as in Part 2) from some skew-symmetric matrixA --------------------------------------------------------------------------- I have solution of question 2) so you don't need to solve 2)and I write down just for your reference I need the solution of 1) and 3) please help! becausetommorrow is my final exam and I only could post 5 question one dayso I could not seperate this question. Let A be a skew-symmetric matrix that AT= -A,assume that A is an nxn matrix 1) show that I+A is invertible 2)show that P=(I-A)(I+A)-1 is orthogonal 3) show that every orthogonal matrix P such that I+P isinvertible arises as in Part 2) from some skew-symmetric matrixA --------------------------------------------------------------------------- I have solution of question 2) so you don't need to solve 2)and I write down just for your reference I need the solution of 1) and 3) please help! becausetommorrow is my final exam and I only could post 5 question one dayso I could not seperate this question.

Explanation / Answer

ask one question in one post. (2) if we can show that PT =P-1 then P will be orthogonal. consider P = ( I- A)( I +A)-1 whereAT= -A PT = { ( I- A)( I +A)-1}T = {(I+A) -1}T(I-A)T         since (AB)T= BT AT ={(I+A)T}-1 (I- AT) = ( I+AT) -1( I + A) since AT =-A               =(I - A)-1(I+A)                                              on the other hand, P-1 = { (I-A)(I+A)-1}-1                                   ={(I+A) -1} -1 ( I-A) -1 = (I- A) -1(I+A) lhs = rhs , so, the equality holds in the givenstatement.

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.