1.

Show that `Delta =Delta_(1)`, where `Delta = |[Ax,x^(2), 1],[By, y^(2), 1],[Cz, z^(2),1]| "and "Delta_(1) = |[A,B, C],[x, y, z],[zy, zx,xy]|`

Answer» We have,
`Delta = |[Ax,x^(2), 1],[By, y^(2), 1],[Cz, z^(2),1]|`
`= |[Ax,By, Cz],[x^(2), y^(2), z^(2)],[1,1 ,1]| ["interchanging the rows and columns"]`
`=(xyz) |[A,B, C],[x, y, z],[(1)/(x),(1)/(y) ,(1)/(z)]| ["applying" C_(1) to (1)/(x) C_(1), C_(2) to (1)/(y) C_(2), C_(3) to (1)/(z)C_(3) " and multiplying the whole determinant by xyz"]`
`=((xyz))/((xyz))* |[A,B, C],[x, y, z],[yz,zx ,xy]| ["applying" R_(3) to (xyz)R_(3)" and dividing the whole determinant by xyz"]`
`=|[A,B, C],[x, y, z],[zy,zx ,xy]| =Delta_(1)`
Hence, `Delta = Delta_(1)`


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