1.

`|(y+z,z,y),(z,z+x,x),(y,x,x+y)|-4xyz`

Answer» Given determinant `=|{:(y+z, " "z, y),(" "z, z+x,x), (" "y, " "x, x+y):}|`
`=|{:(0, -2x, -2x),(z, z+x," "x), (y, " "x, x+y):}| [R_(1) to R_(1) -(R_(2) + R_(3))]`
`=(-2x)*|{:(0, 1, 1),(z, z+x," "x), (y, " "x, x+y):}| ["taking(-2x) common from"R_(1)]`
`=(-2x)*|{:(0, 0, " "1),(z, z," "x), (y, -y, x+y):}| [C_(2) to (C_(2) -C_(3))]`
`=(-2x)*1*|{:(z,z),(y,-y):}| ["expanded by"R_(1)]`d
`=(-2x)*1*(-yz-yz) =(-2x)(-2yz) = 4xyz.`


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