1.

Show that: `|3a-a+b-a+c-b+a3b-b+c-c+a-c+b3c|=3(a+b+c)(a b+b c+c a)dot`

Answer» `" Let " Delta =|{:(3a,,-a+b,,-a+c),(-b+a,,3b,,-b+c),(-c+a,,-c+b,,3c):}|`
`=(a+b+c) |{:(1,,-a+b,,-a+c),(1,,3b,,-b+c),(1,,-c+b,,3c):}|`
`=(a+b+c) |{:(1,,-a+b,,-a+c),(0,,2b+a,,-b+a),(0,,-c+a,,2c+a):}|`
`"[Applyint " R_(2) toR_(2) -R_(1) ,R_(3) to R_(3) -R_(1)]`
`=(a+b+c) |{:(2b+a,,-b+a),(-c+a,,2c+a):}|` [Expanding along `C_(1)]`
`=(a+b+c){(2b+a)(2c+a)-(-b+a)(-c+a)}`
`=(a+b+c){(4bc+2ab+2ca+a^(2))`
`-(bc-ab-ac+a^(2))`
`=(a+b+c)(3bc+3ab+3ca)`
`=3(a+b+c)(ab+bc+ca)`


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