Ex 3.3, 10 (ii) - Chapter 3 Class 12 Matrices (Term 1)

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Ex 3.3, 10 Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (ii) [■8(6&−2&[email protected]−2&3&−[email protected]&−1&3)] Let A = [■8(6&−2&[email protected]−2&3&−[email protected]&−1&3)] A’ = [■8(6&−2&[email protected]−2&3&−[email protected]&−1&3)] 1/2 (A + A’) = 1/2 ([■8(6&−2&[email protected]−2&3&−[email protected]&−1&3)]" + " [■8(6&−2&[email protected]−2&3&−[email protected]&−1&3)]) = 1/2 [■8(12&−4&[email protected]−4&6&−[email protected]&−2&6)] = [■8(6&−2&[email protected]−2&3&−[email protected]&−1&3)] 1/2 (A − A’) = 1/2 ([■8(6&−2&[email protected]−2&3&−[email protected]&−1&3)]−[■8(6&−2&[email protected]−2&3&−[email protected]&−1&3)]) = 1/2 [■8(0&0&[email protected]&0&[email protected]&0&0)] = [■8(0&0&[email protected]&0&[email protected]&0&0)] Let, P = 𝟏/𝟐 (A + A’) = [■8(6&−2&[email protected]−2&3&−[email protected]&−1&3)] P’ = [■8(6&−2&[email protected]−2&3&−[email protected]&−1&3)] = P Since P = P’ P is a symmetric matrix. Let, Q = 𝟏/𝟐 (A − A’) = [■8(0&0&[email protected]&0&[email protected]&0&0)] Q’ = [■8(0&0&[email protected]&0&[email protected]&0&0)] = −[■8(0&0&[email protected]&0&[email protected]&0&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