1.

Solve the determinant using properties`|[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]| = (a+b+c)^3`

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


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