1.

Prove that ` /_ |[a+bx, c+dx, p+qx],[-ax+b, cx+d, px+q],[u,v,w]|=(1-x^2) [[a,c,p],[b,d,q],[u,v,w]] `

Answer» We have
`LHS=|[a+bx, c+dx, p+qx],[ax+b,cx+d,px+q],[u, v, w]|`
`=|[a-ax^(2), c-cx^(2), p-px^(2)],[ax+b,cx+d,px+q],[u, v, w]| ["applying"R_(1) to R_(1) -xR_(2)]`
`=|[a(1-x^(2)), c(1-x^(2)), p(1-x^(2))],[ax+b,cx+d,px+q],[u, v, w]|`
`=(1-x^(2))|[a, c, p],[ax+b,cx+d,px+q],[u, v, w]| ["taking out "(1-x^(2))"common from"R_(1)]`
`=(1-x^(2))|[a, c, p],[b, d, q],[u, v, w]| ["applying "R_(2) to R_(2) -xR_(1)]`
=RHS.
Hence, LHS = RHS.


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