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`Det[[a^2,bc,ac+c^2],[a^2+ab,b^2,ac],[ab,b^2,c^2]]=4a^2b^2c^2` |
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Answer» `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]|` Now expanding the determinant, `=-2a^2b^2c^2(1(-1)-1(1)+0)` `=-2a^2b^2c^2(-2)` `=4a^2b^2c^2=R.H.S.` |
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