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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Example 22 Express the matrix B = [■8(2&−2&−4@−1&3&4@1&−2&−3)] as the sum of a symmetric and a skew symmetric matrix. B = [■8(2&−2&−4@−1&3&4@1&−2&−3)] B’ = [■8(2&−1&1@−2&3&−2@−4&4&−3)] Finding 𝟏/𝟐 (B + B’) and 𝟏/𝟐 (B − B’) 𝟏/𝟐 (B + B’) = 1/2 ([■8(2&−2&−4@−1&3&4@1&−2&−3)]+[■8(2&−1&1@−2&3&−2@−4&4&−3)]) = 1/2 [■8(4&−3&−3@−3&6&2@−3&2&−6)] = [■8(𝟐&(−𝟑)/𝟐&(−𝟑)/𝟐@(−𝟑)/𝟐&𝟑&𝟏@(−𝟑)/𝟐&𝟏&−𝟑)] 𝟏/𝟐 (B – B’) = 1/2 ([■8(2&−2&−4@−1&3&4@1&−2&−3)]−[■8(2&−1&1@−2&3&−2@−4&4&−3)]) = 1/2 [■8(0&−1&−5@1&0&6@5&−6&0)] = [■8(𝟎&(−𝟏)/𝟐&(−𝟓)/𝟐@𝟏/𝟐&𝟎&𝟑@𝟓/𝟐&−𝟑&𝟎)] Let, P = 𝟏/𝟐 (B + B’) = [■8(2&(−3)/2&(−3)/2@(−3)/2&3&1@(−3)/2&1&−3)] P’ = [■8(2&(−3)/2&(−3)/2@(−3)/2&3&1@(−3)/2&1&−3)] = P Since P’ = P P is a symmetric matrix. Let, Q = 𝟏/𝟐 (B − B’) = [■8(0&(−1)/2&(−5)/2@1/2&0&3@5/2&−3&0)] Q’ = [■8(0&1/2&5/2@(−1)/2&0&−3@(−5)/2&3&0)] = – [■8(0&(−1)/2&(−5)/2@1/2&0&3@5/2&−3&0)] Since Q’ = − Q Q is a skew symmetric matrix. Now, P + Q = 1/2 (B + B’) + 1/2 (B − B’) = B Thus, B is a sum of symmetric & skew symmetric matrix

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.