Using cofactors of elements of the third colu | vedclass

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(A) The given determinant is $\Delta = \left| \begin{array}{ccc} 1 & x & yz \ 1 & y & zx \ 1 & z & xy \end{array} ight|$.The elements of the third

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$(x-y)(y-z)(z-x)$

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(A) The given determinant is $\Delta = \left| \begin{array}{ccc} 1 & x & yz \ 1 & y & zx \ 1 & z & xy \end{array} ight|$.The elements of the third column are $a_{13} = yz$,$a_{23} = zx$,and $a_{33} = xy$.The minors are:$M_{13} = \left| \begin{array}{cc} 1 & y \ 1 & z \end{array} ight| = z-y$$M_{23} = \left| \begin{array}{cc} 1 & x \ 1 & z \end{array} ight| = z-x$$M_{33} = \left| \begin{array}{cc} 1 & x \ 1 & y \end{array} ight| = y-x$The cofactors are:$A_{13} = (-1)^{1+3} M_{13} = z-y$$A_{23} = (-1)^{2+3} M_{23} = -(z-x) = x-z$$A_{33} = (-1)^{3+3} M_{33} = y-x$Expanding along the third column:$\Delta = a_{13} A_{13} + a_{23} A_{23} + a_{33} A_{33}$$= yz(z-y) + zx(x-z) + xy(y-x)$$= yz^2 - y^2z + zx^2 - z^2x + xy^2 - x^2y$$= (zx^2 - x^2y) + (xy^2 - y^2z) + (yz^2 - z^2x)$$= x^2(z-y) + y^2(x-z) + z^2(y-x)$$= -(x-y)(y-z)(z-x)$Wait,let us re-evaluate the expansion:$= yz^2 - y^2z + zx^2 - z^2x + xy^2 - x^2y$$= -x^2(y-z) + x(y^2-z^2) - yz(y-z)$$= -(y-z) [x^2 - x(y+z) + yz]$$= -(y-z) [x^2 - xy - xz + yz]$$= -(y-z) [x(x-y) - z(x-y)]$$= -(y-z)(x-y)(x-z)$$= (x-y)(y-z)(z-x)$Thus,$\Delta = (x-y)(y-z)(z-x)$.

Using cofactors of elements of the third column,evaluate $\Delta = \left| \begin{array}{ccc} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{array} \right|$.

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