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

Explanation

We have to proof: `|(a+x, y, z),( x, a+y, z),( x, y, a+z)|=a^2(a+x+y+z)`

Then let

`Delta=|(a+x, y, z),( x, a+y, z),( x, y, a+z)|`

Now use `C_1->C_1+C_2+C_3`

`Delta=|(a+x+y+z, y, z),(a+x+y+z, a+y, z),(a+x+y+z, y, a+z)|`

`implies Delta=(a+x+y+z)|(1, y, z),( 1, a+y, z),( 1, y, a+z)|`

Now use `R_2->R_2-R_1,R_3->R_3-R_1`

`implies Delta=(a+x+y+z)|(1, y, z),( 0, a, 0),( 0, 0, a)|`

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