Question Number 224989 by CrispyXYZ last updated on 15/Oct/25
$$\mathrm{Calculate} \\ $$$${D}_{{n}} =\begin{vmatrix}{{x}_{\mathrm{1}} +{a}_{\mathrm{1}} ^{\mathrm{2}} }&{{a}_{\mathrm{1}} {a}_{\mathrm{2}} }&{\ldots}&{{a}_{\mathrm{1}} {a}_{{n}} }\\{{a}_{\mathrm{2}} {a}_{\mathrm{1}} }&{{x}_{\mathrm{2}} +{a}_{\mathrm{2}} ^{\mathrm{2}} }&{\ldots}&{{a}_{\mathrm{2}} {a}_{{n}} }\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{{a}_{{n}} {a}_{\mathrm{1}} }&{{a}_{{n}} {a}_{\mathrm{2}} }&{\ldots}&{{x}_{{n}} +{a}_{{n}} ^{\mathrm{2}} }\end{vmatrix} \\ $$
Answered by Tinku Tara last updated on 17/Oct/25
$$\mathrm{Matrix}={X}+{aa}^{{T}} \\ $$$${where}\:{X}=\:\mathrm{diag}\left({x}_{\mathrm{1}} ,….,{x}_{{n}} \right)\:+{aa}^{{T}} \:\:\: \\ $$$$\:\:\:\:\:{and}\:{a}=\left({a}_{\mathrm{1}} ,…,{a}_{{n}} \right)^{{T}} \\ $$$${det}\left({X}+{aa}^{{T}} \right)={det}\left({X}\right)\left(\mathrm{1}+{a}^{{T}} {X}^{−\mathrm{1}} {a}\right) \\ $$$${a}^{{T}} {X}^{−\mathrm{1}} {a}=\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{a}_{{i}} ^{\mathrm{2}} }{{x}_{{i}} } \\ $$$${D}_{{n}} =\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}{x}_{{i}} \left(\mathrm{1}+\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{a}_{{i}} ^{\mathrm{2}} }{{x}_{{i}} }\right) \\ $$
Commented by fantastic last updated on 17/Oct/25
$${seeing}\:{you}\:{active}\:{after}\:{a}\: \\ $$$${long}\:{time} \\ $$