Check sibling questions

Example 22 - Express matrix B as sum of symmetric and skew

Example 22 - Chapter 3 Class 12 Matrices - Part 2
Example 22 - Chapter 3 Class 12 Matrices - Part 3 Example 22 - Chapter 3 Class 12 Matrices - Part 4

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Transcript

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