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Prove that |[a^2+1,ab,ac],[ab,b^2+1,bc],...

Prove that `|[a^2+1,ab,ac],[ab,b^2+1,bc],[ac,bc,c^2+1]|=1+a^2+b^2+c^2`

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`|{:(a^(2)+1,,ab,,ac),(ab,,b^(2)+1,,bc),(ac,,bc,,c^(2)+1):}|`
`=(1)/(abc) |{:(a(a^(2)+1),,ab^(2),,ac^(2)),(a^(2)b,,b(b^(2)+1),,bc^(2)),(a^(2),,b^(2),,c(c^(2)+1)):}|`
[Multiplying `C_(1)C_(2)C_(3)` by a,b,c respectively]
`=(abc)/(abc) |{:(a^(2)+1,,b^(2),,c^(2)),(a^(2),,b^(2)+1,,c^(2)),(a^(2),,b^(2),,c^(2)+1):}|`
[Taking common a,b , c from `R_(1)R_(2)R)_(3)` respectively]
`|{:(1+a^(2)+b^(2)+c^(2),,b^(2),,c^(2)),(1+a^(2)+b^(2)+c^(2),,b^(2)+1,,c^(2)),(1+a^(2)+b^(2)+c^(2),,b^(2),,c^(2)+1):}|`
`[C_(1) to C_(1) +C_(2) +C_(3)]`
`= (1+a^(2) +b^(2) +c^(2)) |{:(1,,b^(2),,c^(2)),(1,,b^(2)+1,,c^(2)),(1,,b^(2),,c^(2)+1):}|`
`=(1+a^(2)+b^(2)+c^(2)) |{:(1,,b^(2),,c^(2)),(0,,1,,0),(0,,0,,1):}|`
[Applying `R_(2) to R_(2) -R_(1) " and " R_(3) to R_(3) to R_(3)-R_(1)]`
`=(1+a^(2) +b^(2)+c^(2))`
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