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In a calculation I need the determinant of $\mathbf{I}_n+(a_ia_j)_{1\leq i,j\leq n}$ for given real numbers $a_1,\ldots,a_n$. I conjecture that the determinant equals $1+\sum_{k=1}^na_k^2$, which seems reasonable after making calculations for $n$ up to $5$. However, I did not manage to proof this formula. I tried proving the formula with induction, but any way I tried working out the induction step using linearity of the first row/column did not work. I'm just looking for some help towards a proof, so an inductive proof is not necessary.

Any help is much appreciated.

Václav Mordvinov
  • 3,268
  • It is an immediate consequence of https://en.wikipedia.org/wiki/Sylvester%27s_determinant_identity, compare also https://math.stackexchange.com/questions/420198/proving-the-relation-deti-xyt-1-xty. – Martin R Dec 02 '18 at 17:31
  • Thank you for your reference @MartinR, and sorry for posting the duplicate! I tried searching but did not find the linked questions myself – Václav Mordvinov Dec 02 '18 at 17:33
  • Note that this matrix can be easily inverted, which was not noted in the duplicate (and not asked here in fact) – Damien Dec 02 '18 at 17:57

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