We first characterize thed×dmatrix-valued weightsWon the complex plane ℂ such that the Fock projectionP_αis bounded on the vector-valued spacesL²_α,W(ℂ^d) induced byW. It is proved thatP_αis bounded onL²_α,W(ℂ^d) if and only ifWsatisfies a restrictedA₂-condition. Then we establish some function-theoretic and operator-theoretic properties for the Fock spacesF²_α,W(ℂ^d) induced by thed×dmatrix-valued weightsWsatisfying the restrictedA₂-condition, including the density of polynomials, Littlewood-Paley type estimates, embedding theorems and boundedness of Volterra type integration operators induced byd×dmatrix-valued entire functions.