Prove that `|1+a1 1 1 1 1+b1 1 1 1 1+c1 1 1 1 1+d|=a b c d(a+1/a+1/b+1/c+1/d)dot` Hence find the value of the determinant if `a ,b ,c , d` are the roots of the equation `p x^4+q x^3+r x^2+s x+t=0.`

Answer

`(t-s)/(p)`

Explanation

Applying
`R_(1) to (1)/(a)R_(1),R_(2) to (1)/(b) R_(2),R_(3)to (1)/(c ) R_(3),R_(4) to (1)/(d) R_(4)`
we get
`Delta = abcd |{:(1+(1)/(a),,(1)/(a),,(1)/(a),,(1)/(a)),((1)/(b),,1+(1)/(b),,(1)/(b),,(1)/(b)),((1)/(c ),,(1)/(c ),,1+(1)/(c ),,(1)/(c )),((1)/(d),,(1)/(d),,(1)/(d),,1+(1)/(d)):}|`
Applying `R_(1) to R_(1)+R_(2)+R_(3)+R_(4)` and taking `1+(1)/(a)+(1)/(b)+(1)/(c)+(1)/(d)` common we get
` Detla =abcd (1+(1)/(a)+(1)/(b)+(1)/(c )+(1)/(d))xx |{:(1,,1,,1,,1),((1)/(b),,1+(1)/(b),,(1)/(b),,(1)/(b)),((1)/(c ),,(1)/(c ),,1+(1)/(c ),,(1)/(c )),((1)/(d),,(1)/(d),,(1)/(d),,1+(1)/(d)):}|`
Now applying `C_(2) to C_(2)-C_(1),C_(3) to C_(3) -C_(1),C_(4)to C_(4)-C_(1)`
`Delta =abcd (1+(1)/(a)+(1)/(b)+(1)/(c)+(1)/(d))xx |{:(1,,0,,0,,0),((1)/(b),,1,,0,,0),((1)/(c ),,0,,1,,0),((1)/(d),,0,,0,,1):}|`
Expanding along `R_(1)` we get
`Delta= abcd (1+(1)/(a)+(1)/(b)+(1)/(c)+(1)/(d)) xx |{:(1,,0,,0),(0,,1,,0),(0,,0,,1):}|`
Expanding again along `R_(1)` we get
`Delta =abcd (1+(1)/(a)+(1)/(b)+(1)/(c)+(1)/(d))`
2nd Part: `Delta =abcd +(bcd+acd+abd+abc)`
`=(t)/(p)-(s)/(p)=(t-s)/(p)`