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Question 1 a) Let |-1 2 5) A = 1 0 -3,B = 4 5,U= 5, V = (2 -3). 16 -4 2/ /1-3 [2) 0 6/ Calculate the quantities i. AB-UV, (A-21)U. ii. U'U, UU". iii. det A,det (B'B). b) A square nxn matrix S is called symmetric, if S' = S, and it is called anti- symmetric if S' = -S. Any square nxn matrix A can be written as the sum of a symmetric matrix S and an anti-symmetric matrix N, of the same dimensions, by setting: S: = N= so that A = S+ N. i. Show that S is indeed symmetric (i.e. S = S'), that N is indeed antisymmetric (i.e. N' = -N) and that A = S+ N. ii. Write the matrix A of question (1.a) as the sum of a symmetric and an antisymmetric matrix (i.e. calculate S and N).

Question 1 a) Let |-1 2 5) A = 1 0 -3,B = 4 5,U= 5, V = (2 -3). 16 -4 2/ /1-3 [2) 0 6/ Calculate the quantities i. AB-UV, (A-21)U. ii. U'U, UU". iii. det A,det (B'B). b) A square nxn matrix S is called symmetric, if S' = S, and it is called anti- symmetric if S' = -S. Any square nxn matrix A can be written as the sum of a symmetric matrix S and an anti-symmetric matrix N, of the same dimensions, by setting: S: = N= so that A = S+ N. i. Show that S is indeed symmetric (i.e. S = S'), that N is indeed antisymmetric (i.e. N' = -N) and that A = S+ N. ii. Write the matrix A of question (1.a) as the sum of a symmetric and an antisymmetric matrix (i.e. calculate S and N).

Advanced Engineering Mathematics
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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Question 1 a) Let |-1 2 5) A = 1 0 -3,B = 4 5,U= 5, V = (2 -3). 16 -4 2/ /1-3 [2) 0 6/ Calculate the quantities i. AB-UV, (A-21)U. ii. U'U, UU". iii. det A,det (B'B). b) A square nxn matrix S is called symmetric, if S' = S, and it is called anti- symmetric if S' = -S. Any square nxn matrix A can be written as the sum of a symmetric matrix S and an anti-symmetric matrix N, of the same dimensions, by setting: S: = N= so that A = S+ N. i. Show that S is indeed symmetric (i.e. S = S'), that N is indeed antisymmetric (i.e. N' = -N) and that A = S+ N. ii. Write the matrix A of question (1.a) as the sum of a symmetric and an antisymmetric matrix (i.e. calculate S and N).
Transcribed Image Text:Question 1 a) Let |-1 2 5) A = 1 0 -3,B = 4 5,U= 5, V = (2 -3). 16 -4 2/ /1-3 [2) 0 6/ Calculate the quantities i. AB-UV, (A-21)U. ii. U'U, UU". iii. det A,det (B'B). b) A square nxn matrix S is called symmetric, if S' = S, and it is called anti- symmetric if S' = -S. Any square nxn matrix A can be written as the sum of a symmetric matrix S and an anti-symmetric matrix N, of the same dimensions, by setting: S: = N= so that A = S+ N. i. Show that S is indeed symmetric (i.e. S = S'), that N is indeed antisymmetric (i.e. N' = -N) and that A = S+ N. ii. Write the matrix A of question (1.a) as the sum of a symmetric and an antisymmetric matrix (i.e. calculate S and N).
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