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Det[[a^2,bc,ac+c^2],[a^2+ab,b^2,ac],[ab,...

`Det[[a^2,bc,ac+c^2],[a^2+ab,b^2,ac],[ab,b^2,c^2]]=4a^2b^2c^2`

Text Solution

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`L.H.S. = |[a^2,bc,ac+c^2],[a^2+ab,b^2,ac],[ab,b^2+bc,c^2]|`
Applying `R_1->R_1-R_2-R_3`
`= |[-2ab,-2b^2,0],[a^2+ab,b^2,ac],[ab,b^2+bc,c^2]|`
`=-2 |[ab,b^2,0],[a^2+ab,b^2,ac],[ab,b^2+bc,c^2]|`
Now applying `R_2->R_2-R_1,R_3 ->R_3-R_1`
`= -2 |[ab,b^2,0],[a^2,0,ac],[0,bc,c^2]|`
`=-2abc |[a,b,0],[a,0,c],[0,b,c]|`
`=-2a^2b^2c^2 |[1,1,0],[1,0,1],[0,1,1]|`
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