We prove that, for $X$, $Y$, $A$ and $B$ matrices with entries in a non-commutative ring such that $$ \\hbox{$X{ij},Y{k\\ell}=-A{i\\ell} B{kj}$}, $$ satisfying suitable commutation relations (in particular, $X$ is a Manin matrix), row-pseudo-commutative matrix (a Manin matrix), the following identity holds: $$ \\mathrm {col-det } \\ X \\ \\mathrm { col-det } \\ Y \\ = \\langle 0\\mid \\mathrm { col-det } \\ (aA + X (I-a^{\\dagger} B)^{-1} Y)\\mid 0\\rangle $$ Furthermore, if also $Y$ is a Manin matrix, $Y{ij},Y{kl}=0$ for $i\\neq k$, $j\\neq l$ $$ \\mathrm {col-det } \\ X \\ \\mathrm { col-det } \\ Y =\\int \\mathcal{D}(\\psi, \\bar{\\psi}) \\exp \\big(\\sum\_{k \\geq 0}\\frac{(\\bar{\\psi} A \\psi)^{k}}{k+1}(\\bar{\\psi} X B^k Y \\psi)\\big) $$ Here $\\langle 0 \\mid$ and $\\mid 0\\rangle$, are respectively the bra and the ket of the ground state, $a^{\\dagger}$ and $a$ the creation and annihilation operators of a quantum harmonic oscillator, while $\\bar{\\psi}i$ and $\\psi_i$ are Grassmann variables in a Berezin integral. These results should be seen as a generalization of the classical Cauchy–Binet formula, in which $A$ and $B$ are null matrices, and of the non-commutative generalization, the Capelli identity, in which $A$ and $B$ are identity matrices and $\[X{ij},X{k\\ell}]=\[Y{ij},Y\_{k\\ell}]=0$.