1.

Using properties of determinants, prove that`|(a+x, y, z),( x, a+y, z),( x, y, a+z)|=a^2(a+x+y+z)`

Answer» Let the value of the given determinant be `Delta`. Then.
`Delta =|{:(a+x, " "y, " "z), (" "x, a+y, " "z), (" "x, " "y, a+z):}|`
`=|{:(a+x+y+z, " "y, " "z), (a+x+y+z, a+y, " "z), (a+x+y+z, " "y, a+z):}| " "[C_(1) to C_(1) + C_(2) + C_(3)]`
`=(a +x +y+z)*|{:(1, y, z), (1, a+y, z), (1, y, a+z):}| " "["taking (a+x+y+z) common from"C_(1)]`
`=(a +x +y+z)*|{:(1, y, z), (0, a, 0), (0, 0, a):}| " "[R_(2) to R_(2) - R_(1) " and"R_(3) to R_(3)-R_(1)]`
`=(a +x +y+z)*1*|{:(a, 0), (0,a):}| " "["expanded by"C_(1)]`
` =a^(2) (a+x+y+z).`
Hence, `Delta =a^(2) (a+x+y+z)`


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