[Question 2014a02] solution 3 (descending order formula)

[Question 2014a02] solution 3 (descending order formula)

Write the matrix \ (A \) as follows:

\ [A =\begin {pmatrix}-2A_1 & 0 & \ cdots & 0 & 0 \ 0 &-2a_2 & \ cdots & 0 & 0 \ vdots & \ vdots & \ vdots \ 0 & 0 & \ cdots &-2a _ {n-1} & 0 \ 0 & 0 & \ cdots & 0 &-2a_n \ end {pmatrix} \]

\ [+ \ Begin {pmatrix} A_1 & 1 \ A_2 & 1 \ vdots & \ vdots \ A _ {n-1} & 1 \ a_n & 1 \ end {pmatrix} \ cdot I _2 ^ {-1} \ cdot \ begin {pmatrix} 1 & 1 & \ cdots & 1 & 1 \ A_1 & A_2 & \ cdots & A _ {n-1} & a_n \ end {pmatrix }. \]

Available from the descending order Formula

\ [| A | =\begin {vmatrix}-2A_1 & 0 & \ cdots & 0 & 0 \ 0 &-2a_2 & \ cdots & 0 & 0 \ vdots & \ vdots & \ vdots \ 0 & 0 & \ cdots &-2a _ {n-1} & 0 \ 0 & 0 & \ cdots & 0 &- 2a_n \ end {vmatrix} \ cdot \ Bigg | I _2 + \ begin {pmatrix} 1 & 1 & \ cdots & 1 & 1 \ A_1 & A_2 & \ cdots & _{ n-1} & a_n \ end {pmatrix} \ begin {pmatrix}-2A_1 & 0 & \ cdots & 0 & 0 \ 0 &-2a_2 & \ cdots & 0 & 0 \\ \ vdots & \ vdots \ 0 & \ cdots &-2a _ {n-1} & 0 \ 0 & 0 & \ cdots & 0 &-2a_n \ end {pmatrix} ^ {-1} \ begin {pmatrix} A_1 & 1 \ A_2 & 1 \ vdots & \ vdots \ A _ {n-1} & 1 \ a_n & 1 \ end {pmatrix} \ Bigg | \]

\ [= (-2) ^ n \ prod _ {I = 1} ^ na_ I \ begin {vmatrix} 1-\ frac {n} {2} &-\ frac {1} {2} \ sum _ {I = 1} ^ n \ frac {1} {a_ I} \-\ frac {1} {2} \ sum _ {I = 1} ^ na_ I & 1 -\ frac {n} {2} \ end {vmatrix} \]

\ [= (-2) ^ {N-2} \ prod _ {I = 1} ^ na_ I \ Bigg (n-2) ^ 2-\ big (\ sum _ {I = 1} ^ na_ I \ big) \ big (\ sum _ {I = 1} ^ n \ frac {1} {a_ I} \ big) \ Bigg ). \ quad \ Box \]

[Question 2014a02] solution 3 (descending order formula)