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Prove that: |(a,b-c,c-b),(a-c,b,c-a),(a-...

Prove that: `|(a,b-c,c-b),(a-c,b,c-a),(a-b,b-a,c)|=(a+b-c)(b+c-a)(c+a-b)`

Text Solution

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`[[a+b+c-c-b,b-c,c-b],[a-c+b+c-a,b,c-a],[a-b+b-a+c,b-a,c]]`
`C_2->C_2+C_3`
`[[a,b-c+c-b,c-b],[b,b+c-a,c-a],[c,b-a+c,c]]`
`[[a,0,c-b],[b,b+c-a,c-a],[c,b+c-a,c]]`
`(b+c-a)[[a,0,c-b],[b,1,c-a],[c,1,c]]`
`(b+c-a)[a(c-c+a)+(-b)(b-c)-(b-c)(b-c)]`
`(b+c-a)(a-(b-c))(a+(b-c))`
`(a+b-c)(b+c-a)(a+c-b)`.
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