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Ex 3.3, 10 - Express as sum of a symmetric, a skew symmetric

Ex 3.3, 10 - Chapter 3 Class 12 Matrices - Part 2

 

 


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Ex 3.3, 10 Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (i) [■8(3&[email protected]&−1)] Let A = [■8(3&[email protected]&−1)] A’ = [■8(3&[email protected]&−1)] 1/2 (A + A’) = 1/2 ([■8(3&[email protected]&−1)]+ [■8(3&[email protected]&−1)]) = 1/2 [■8(6&[email protected]&−2)] = [■8(3&[email protected]&−1)] 1/2 (A – A’) = 1/2 ([■8(3&[email protected]&−1)]" − " [■8(3&[email protected]&−1)]) = 1/2 [■8(0&[email protected]−4&0)] = [■8(0&[email protected]−2&0)] Let, P = 𝟏/𝟐 (A + A’) = [■8(3&[email protected]&−1)] P’ = [■8(3&[email protected]&−1)] = P Since P‘ = P P is a symmetric matrix. Let, Q = 𝟏/𝟐 (A − A’) = [■8(0&[email protected]−2&0)] Q’ = [■8(0&−[email protected]&0)] = – [■8(0&[email protected]−2&0)]= −Q Since Q’ = − Q Q is a skew symmetric matrix. Now, P + Q = 1/2 (A + A’) + 1/2 (A − A’) = A Thus, A is a sum of symmetric & skew symmetric matrix

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.