65. Final exam, Winter 2015. A matrix A is said to be skew-symmetric if AT = -A. For instance 03 М- |-3 0 and N = 0. are both skew-symmetric matrices -- -7 (a) Prove that a skew-symmetric matrix with an odd number of rows cannot be invertible. (b) We are given a set of 3 vectors u1, u2, u3 in R®, and we know that they are linearly independent. Further, we have three real numbers a, b, c one of them equal to 0, and we define w1 = -au2+bu3, w2 = au¡ – cu3 and w3 = -bu1+cu2. Are the vectors w1, W2, W3 linearly dependent or not? Justify (If you wish, to answer this question you can use the result of part (a), even if you cannot prove it).
65. Final exam, Winter 2015. A matrix A is said to be skew-symmetric if AT = -A. For instance 03 М- |-3 0 and N = 0. are both skew-symmetric matrices -- -7 (a) Prove that a skew-symmetric matrix with an odd number of rows cannot be invertible. (b) We are given a set of 3 vectors u1, u2, u3 in R®, and we know that they are linearly independent. Further, we have three real numbers a, b, c one of them equal to 0, and we define w1 = -au2+bu3, w2 = au¡ – cu3 and w3 = -bu1+cu2. Are the vectors w1, W2, W3 linearly dependent or not? Justify (If you wish, to answer this question you can use the result of part (a), even if you cannot prove it).
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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Transcribed Image Text:65.
Final exam, Winter 2015. A matrix A is said to be skew-symmetric if AT = -A. For instance
03
М-
|-3 0
and N =
0.
are both skew-symmetric matrices
--
-7
(a) Prove that a skew-symmetric matrix with an odd number of rows cannot be invertible.
(b) We are given a set of 3 vectors u1, u2, u3 in R®, and we know that they are linearly independent. Further,
we have three real numbers a, b, c one of them equal to 0, and we define w1 = -au2+bu3, w2 = au¡ – cu3
and w3 = -bu1+cu2. Are the vectors w1, W2, W3 linearly dependent or not? Justify (If you wish, to answer
this question you can use the result of part (a), even if you cannot prove it).
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