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|[1+a^2-b^2,2ab,-2b],[2ab,1-a^2+b^2,2a],...

`|[1+a^2-b^2,2ab,-2b],[2ab,1-a^2+b^2,2a],[2b,-2a,1-a^2-b^2]|=(1+a^2+b^2)^3`

Text Solution

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Operating `C_(1)to C_(1)-bC_(3),C_(2)toC_(2)+aC_(3)` we get
`Delta= |{:(1+a^(2)+b^(2),,0,,-2b),(0,,1+a^(2)+b^(2),,2a),(b(1+a^(2)+b^(2)),,-a(1+a^(2)+b^(2)),,1-a^(2)-b^(2)):}|`
`=(1+a^(2)+b^(2))^(2)xx |{:(1,,0,,-2b),(0,,1,,2a),(b,,-a,,1-a^(2)-b^(2)):}|`
`R_(3) to R_(3) -bR_(1)+aR_(2)` gives
`Delta =(1+a^(2)+b^(2))^(2)xx |{:(1,,0,,-2b),(0,,1,,2a),(0,,0,,1-a^(2)-b^(2)):}|`
`=(1+a^(2)+b^(2))^(3)" " "[Expanding along "C_(1)]`
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