Ex 3.3

Chapter 3 Class 12 Matrices
Serial order wise

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