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Question
Using Cofactors of elements of third column, evaluate `triangle = |(1,x,yz),(1,y,zx),(1,z,xy)|`
Solution
`A_13 = -1^(1 + 3) abs ((1,y),(1,z)) = (1)(z- y) = (z- y)`
`A_23 = -1^ (2 + 3) abs ((1,x),(1,z)) = (1)(z- x) = - (x - z)`
`A_33 = -1^(1 + 3) abs ((1,x),(1,y)) = (1)(y - x) = (y - x)`
`Delta = a_13 A_13 + a_23 A_23 + a_33A_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 - z^2y + yz^2 - y^2z`
`= x^2 (z - y) + z(y - z) (y + z) + yz (z - y)`
`= (z - y) [x^2 - x(y + z) + yz]`
`= (z - y) [x^2 - xy - xz + yz]`
`= (z - y) [x (x - y) - z (x - y)]`
`= (z - y)(x - y)(x - z)`
`= (x - y)(y - z)(z - x)`
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