Using properties of determinants, prove that `|[a^2, bc, ac+c^2] , [a^2+ab, b^2, ac] , [ab, b^2+bc, c^2]|` = `4a^2b^2c^2`
Using properties of determinants, prove that `|[a^2, bc, ac+c^2] , [a^2+ab, b^2, ac] , [ab, b^2+bc, c^2]|` = `4a^2b^2c^2`
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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)):}|=|{:(-2ab,-2b^(2),ac+c^(2)),(a^(2)+ab,b^(2),ac),(ab,b^(2)+bc,c^(2)):}|`
`(R_(1)toR_(1)-R_(2)-R_(3))`
`=abc|{:(-2b,-2b,0),(a+b,b,a),(b,b+c,c):}|-2ab^(2)|{:(1,1,0),(a+b,b,a),(b,b+c,c):}|`
`|{:(-2b,-2b,0),(a+b,b,a),(b,b+c,c):}|-2ab^(2)|{:(1,1,0),(a+b,b,a),(b,b+c,c):}|" "(C_(2)toC_(2)-C_(1))`
`=-2ab^(2)c.1|{:(-a,a),(c,c):}|=-2ab^(2)c(ac-ac)`
(Expanding along `R_(1)`)
`=4a^(2)b^(2)c^(2)=R.H.S.`
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