# Using Cofactors of elements of third column, evaluate $\Delta = \begin{vmatrix} 1&x&yz \\ 1&y&zx \\ 1&z&xy \end{vmatrix}$

$\begin{array}{1 1} (x+y)(y+z)(z+x) \\ (x-y(z-y)(x-z) \\ (x-y)(y-z)(z-x) \\ (y-z)(y-x)(z-x) \end{array}$

Toolbox:
• $\bigtriangleup$=Sum of the product of any row (or column) with their corresponding cofactors.
• Given:
• Using cofactors of elements of third column evaluate
$\bigtriangleup=\begin{vmatrix}1 & x & yz\\1 & y & zx\\1 & z & xy\end{vmatrix}$
we know that expanding the determinant $\bigtriangleup$ along $C_3$ we have,
$\bigtriangleup=a_{13}A_{13}+a_{23}A_{23}+a_{33}A_{33}$
Minor of$M_{13}=\begin{vmatrix}1 & y\\1 & z\end{vmatrix}=(z-y)$
Minor of$M_{23}=\begin{vmatrix}1 & x\\1 & z\end{vmatrix}=(z-x)$
Minor of$M_{33}=\begin{vmatrix}1 & x\\1 & y\end{vmatrix}=(y-x)$
$A_{13}=(-1)^{1+3}(z-y)=z-y.$
$A_{23}=(-1)^{2+3}(z-x)=-(z-x).$
$A_{33}=(-1)^{3+3}(y-x)=y-x.$
Hence $a_{13}A_{13}+a_{23}A_{23}+a_{33}A_{33}=yz(z-y)+zx(-)(z-x)+xy(y-x)$
$\;\;\;=(z^2y-y^2z)-(zx^2-x^2z)+(xy^2-x^2y)$
$\bigtriangleup=(x^2z-y^2z)+(yz^2-xz^2)+(xy^2-x^2y)$
Taking the common factor
$\bigtriangleup=z(x^2-y^2)+z^2(y-x)+xy(y-x)$
We know $(a^2-b^2)=(a+b)(a-b)$
Therefore $\bigtriangleup=z(x+y)(x-y)+z^2(y-x)+z^2(y-x)+xy(y-x)$
Taking (x-y)as a common factor,
$\bigtriangleup=(x-y)[z(x+y)-z^2-xy]$
$\quad=(x-y)[zx+zy-z^2-xy]$
$\quad=(x-y)[z(x-z)+y(z-x)]$
Taking the common factor x(z-x)
$\bigtriangleup=(x-y)[(z-x)(y-z)]$
$\bigtriangleup=(x-y)(z-x)(y-z)$
Hence $\bigtriangleup=(x-y)(y-z)(z-x)$