11. A matrix ME Mnxn (C) is called skew-symmetric if Mt = -M. Prove that if M is skew-symmetric and n is odd, then M is not invert- ible. What happens if n is even?
11. A matrix ME Mnxn (C) is called skew-symmetric if Mt = -M. Prove that if M is skew-symmetric and n is odd, then M is not invert- ible. What happens if n is even?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please answer 11 and 14. If everything’s good will get thumbs up
![11. A matrix M E Mnxn (C) is called skew-symmetric if Mt = -M.
Prove that if M is skew-symmetric and n is odd, then M is not invert-
ible. What happens if n even?
12. A matrix QE Mnxn (R) is called orthogonal if QQ¹ = I. Prove that
if Q is orthogonal, then det(Q) = ±1.
13. For M€ Mnxn (C), let M be the matrix such that (M)ij = Mij for all
i, j, where Mij is the complex conjugate of Mij.
(a) Prove that det (M) = det (M).
(b)
A matrix Q Mnxn (C) is called unitary if QQ* = = I, where
Q* = Qt. Prove that if Q is a unitary matrix, then | det(Q)] = 1.
=
14. Let 3 {u1, U2,..., un} be a subset of F" containing n distinct vectors,
and let B be the matrix in Mnxn (F) having u, as column j. Prove that
ß is a basis for F" if and only if det (B) 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2F015f3577-1bba-490c-bce6-541838886ae5%2Fdvny9l8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:11. A matrix M E Mnxn (C) is called skew-symmetric if Mt = -M.
Prove that if M is skew-symmetric and n is odd, then M is not invert-
ible. What happens if n even?
12. A matrix QE Mnxn (R) is called orthogonal if QQ¹ = I. Prove that
if Q is orthogonal, then det(Q) = ±1.
13. For M€ Mnxn (C), let M be the matrix such that (M)ij = Mij for all
i, j, where Mij is the complex conjugate of Mij.
(a) Prove that det (M) = det (M).
(b)
A matrix Q Mnxn (C) is called unitary if QQ* = = I, where
Q* = Qt. Prove that if Q is a unitary matrix, then | det(Q)] = 1.
=
14. Let 3 {u1, U2,..., un} be a subset of F" containing n distinct vectors,
and let B be the matrix in Mnxn (F) having u, as column j. Prove that
ß is a basis for F" if and only if det (B) 0.
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