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Using the property of determinants and without expanding, prove that:`|-a^2a b a c b a b^2b cc a c b-c^2|=4a^2b^2c^2`

Text Solution

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`|{:(-a^(2),ab,ac),(ba,-b^(2),ac),(ca,cb,-c^(2)):}|=abc|{:(-a,b,c),(a,-b,c),(a,b,-c):}|`
` =a^(2)b^(2)c^(2)|{:(-1,1,1),(a,-b,c),(a,b,-c):}|`
`=a^(2)b^(2)c^(2)|{:(-1,1,1),(0,-1,1),(2,1,-1):}|(C_(1)toC_(1)+C_(2))`
`=a^(2)b^(2)c^(2).2|{:(1,1),(-1,1):}|`
(Expanding along `C_(1)`)
`=a^(2)b^(2)c^(2)(1+1)=4a^(2)b^(2)c^(2)=R.H.S.`
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