Let A and B be two n xn symmetric matrices. (a) Which of the following is an example that shows that the product AB is not necessarily symmetric? -1 0 O AB = 0 0 O 1 -1 1 O AB = 1 0 1 0 0 1 1 1 O AB = 1 1 0 O AB = 0 0 1 0 O AB = 1 0 (b) Prove that the product AB is symmetric if and only if AB = BA. Remember that A and B are two n xn symmetric matrices. To prove that AB is symmetric, you need to show that it is equal to its transpose, (AB)'. Which of the following proves that if AB = BA then AB is symmetric? (AB)" O (AB)T = BTAT = BA = AB = (BA)2T = BA = AB %3D O (AB)" = (AB)27 = AB O (AB)T = ATBT = BA = AB %3D Which of the following proves that if AB is symmetric then AB = BA? O AB = (AB)T = (BA)2T = BA AB = (BA) = (AB)2T = BA AB = (AB)T = BTAT = BA AB = (AB)T = ATBT = BA O O O O
Let A and B be two n xn symmetric matrices. (a) Which of the following is an example that shows that the product AB is not necessarily symmetric? -1 0 O AB = 0 0 O 1 -1 1 O AB = 1 0 1 0 0 1 1 1 O AB = 1 1 0 O AB = 0 0 1 0 O AB = 1 0 (b) Prove that the product AB is symmetric if and only if AB = BA. Remember that A and B are two n xn symmetric matrices. To prove that AB is symmetric, you need to show that it is equal to its transpose, (AB)'. Which of the following proves that if AB = BA then AB is symmetric? (AB)" O (AB)T = BTAT = BA = AB = (BA)2T = BA = AB %3D O (AB)" = (AB)27 = AB O (AB)T = ATBT = BA = AB %3D Which of the following proves that if AB is symmetric then AB = BA? O AB = (AB)T = (BA)2T = BA AB = (BA) = (AB)2T = BA AB = (AB)T = BTAT = BA AB = (AB)T = ATBT = BA O O O O
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:Let A and B be two n xn symmetric matrices.
(a) Which of the following is an example that shows that the product AB is not necessarily symmetric?
-1 0
O AB =
0 0
0 1
-1 1
O AB =
1 0
1 0
0 1
1 1
O AB =
1
1 0
O AB =
0 0
1 0
1 0
O AB =
1 0
(b) Prove that the product AB is symmetric if and only if AB = BA. Remember that A and B are two n xn symmetric matrices.
To prove that AB is symmetric, you need to show that it is equal to its transpose, (AB)'. Which of the following proves that if AB = BA then AB is symmetric?
O (AB)"
= (BA)2T = BA = AB
%3D
O (AB)T = BTAT = BA = AB
O (AB)T = (AB)2T = AB
O (AB)T = ATBT = BA = AB
%3D
Which of the following proves that if AB is symmetric then AB = BA?
O AB = (AB)T = (BA)²T = BA
AB = (BA)T = (AB)2T = BA
AB = (AB)T = BTAT = BA
AB = (AB)T = ATBT = BA
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