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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).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.5: Iterative Methods For Computing Eigenvalues
Problem 51EQ
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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).
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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