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Prove that : (i) `|{:(a,c,a+c),(a+b,b,a),(b,b+c,c):}|=2 abc` (ii) Prove that : `|{:(a^(2),bc,ac+c^(2)),(a^(2)+ab,b^(2),ac),(ab,b^(2)+bc,c^(2)):}|=4a^(2)b^(2)c^(2)`

Answer» `|{:(a^(2), bc, c^(2)+ac), (a^(2)+ab, b^(2), ac), (ab, b^(2)+bc, c^(2)):}|`
`=(abc)*|{:(" "a, " "c, ca), (a+b, " "b, a), (" "b, b+c, c):}|`
`["taking a, b, c common from"C_(1), C_(2) " and "C_(3) "respectively"]`
`=(abc)*|{:(a, " "c, c+a), (0, -2c, -2c), (b, b+c, " "c):}| " "[R_(2)to R_(2) -(R_(1) + R_(3))]`
`=(abc)*|{:(a, -a, c+a), (0, 0, -2c), (b, b, c):}| " "[C_(2)to C_(2) -C_(3)]`
`=(abc)(2c)*|{:(a, -a), (b, b):}| " "["expanded by"R_(2)]`
` = 2abc^(2) * (ab+ab) = 2abc^(2)(2ab)`
`=4a^(2)b^(2)c^(2)`.


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