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Using cofactors of elements of third column, evaluate \(\Delta = \left| {\begin{array}{*{20}{c}} 1&x&{yz} \\ 1&y&{zx} \\ 1&z&{xy} \end{array}} \right|\)
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MathsClass 12

Using cofactors of elements of third column, evaluate \(\Delta = \left| {\begin{array}{*{20}{c}} 1&x&{yz} \\ 1&y&{zx} \\ 1&z&{xy} \end{array}} \right|\)

Answer

\(\Delta\) = a13A13 + a23A23 + a33A33
= yz( z - y) + zx( x - z) + xy( y - x)
= yz2 - y2z + zx2 - z2x + xy2 - x2y
= zx2 - x2y + xy2 - z2x + yz2 - y2z
= x2 ( z- y) + x(y2 - z2) + yz(z - y)  
= (z - y)[ x2 - x( z + y) + yz]
= (z - y)[x2 - xz - xy +yz]
= (z - y) [ x(x - y) - z(x - y)]
= (z - y)[ ( x - y)(x - z)]
= (z - y)( x - y) (x - z)
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