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Without expanding the determinant, prove...

Without expanding the determinant, prove that `|(a,a^2,bc),(b,b^2,ca),(c,c^2,ab)|=|(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3)|`

Text Solution

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`L.H.S. = |[a,a^2,bc],[b,b^2,ca],[c,c^2,ab]|`
Multiplying first row with `a`, second row with `b` and third row with `c`
`= 1/(abc)|[a^2,a^3,abc],[b^2,b^3,cab],[c^2,c^3,abc]|`
Now, taking, `abc` common in `C_3`
`= (abc)/(abc)|[a^2,a^3,1],[b^2,b^3,1],[c^2,c^3,1]|`
`= |[a^2,a^3,1],[b^2,b^3,1],[c^2,c^3,1]|`
We know, when we interchange any two rows or columns, sign of dererminant changes.
`:.` Interchanging `C_1` and `C_3`
`=(-1)|[1,a^3,a^2],[1,b^3,b^2],[1,c^3,c^2]|`
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