Example 22 - Chapter 3 Class 12 Matrices (Term 1)

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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