1.

Without expanding prove that `|[x+y ,z,1],[y+z,x,1],[z+x,y,1]|=0`

Answer» The given determinant
`=|{:(x+y, y+z, z+x), (z, x, y), (1, 1, 1):}|`
`=|{:(x+y+z, x+y+z, x+y+z), (z, x, y), (1, 1, 1):}| [R_(1) to R_(1) + R_(2)]`
`=(x+y+z)*|{:(1, 1, 1), (z, x, y), (1, 1, 1):}| ["taking" (x+y+z) "common from "R_(1)]`
` = (x+y+z) xx 0 = 0 " "[because R_(1) " and "R_(3) " are identical"].`


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