InterviewSolution
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InterviewSolution
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Prove that `|{:(b+c, c+a, a+b),(c+a, a+b,b+c),(a+b, b+c, c+a):}| =2(a+b+c)(ab+bc+ca-a^(2)-b^(2)-c^(2)).` |
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Answer» Let the given determinant be `Delta`. Then, `Delta =|{:(b+c, c+a, a+b),(c+a, a+b,b+c),(a+b, b+c, c+a):}| ` `=|{:(2(a+b+c), 2(a+b+c), 2(a+b+c)),(c+a, a+b,b+c),(a+b, b+c, c+a):}| [R_(1) to (R_(1) + R_(2) + R_(3))] ` `=2(a+b+c)*|{:(" "1, " "1, " "1),(c+a, a+b,b+c),(a+b, b+c, c+a):}| ["taking(a+b+c)common from"R_(1)]` `=2(a+b+c)*|{:(" "1, " "0, " "0),(c+a, b-c,b-a),(a+b, c-a, c-b):}| [{:(C_(2) to (C_(2)-C_(1))" and"),(C_(3) to (C_(3) -C_(1))):}]` `=2(a+b+c)*1*|{:(b-c, b-a), (c-a, c-b):}| ["expanded by"R_(1)]` `2(a+b+c) *[(b-c)(c-b)-(c-a)(b-a)]` `=2(a+b+c)(ab+bc+ca-a^(2)-b^(2)-c^(2))`. Hence, `Delta =2(a+b+c)(ab+bc+ca-a^(2)-b^(2)-c^(2))`. |
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