On Zagier's Conjecture About the Inverse of a Matrix Related to Double Zeta Values

We prove a conjecture of Zagier about the inverse of a $(K-1)\times (K-1)$ matrix $A=A_K$ using elementary methods. This formula allows one to express the the product of single zeta values $\zeta (2r)\zeta (2K+1-2r)$, $1\leq r\leq K-1$,intermsofthedoublezetavalues$\zeta(2r,2K+1-2r)$,$1\leq r\leq K-1$and$\zeta(2K+1)$.