Symmetric and skew symmetric matrices

Chapter 3 Class 12 Matrices
Concept wise

### Transcript

Ex 3.3, 10 Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (i) [■8(3&5@1&−1)] Let A = [■8(3&5@1&−1)] A’ = [■8(3&1@5&−1)] 𝟏/𝟐 (A + A’) = 1/2 ([■8(3&5@1&−1)]+ [■8(3&1@5&−1)]) = 1/2 [■8(6&6@6&−2)] = [■8(𝟑&𝟑@𝟑&−𝟏)] 𝟏/𝟐 (A – A’) = 1/2 ([■8(3&5@1&−1)]" − " [■8(3&1@5&−1)]) = 1/2 [■8(0&4@−4&0)] = [■8(𝟎&𝟐@−𝟐&𝟎)] Let, P = 𝟏/𝟐 (A + A’) = [■8(3&3@3&−1)] P’ = [■8(3&3@3&−1)] = P Since P‘ = P P is a symmetric matrix. Let, Q = 𝟏/𝟐 (A − A’) = [■8(0&2@−2&0)] Q’ = [■8(0&−2@2&0)] = – [■8(0&2@−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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.