G.-C. Rota [Z. Wahrscheinlichkeitstheor. Verw. Geb. 2, 340–368 (1964; Zbl 0121.02406)] extended the Möbius inversion technique to partially ordered set (posets), provided they are locally finite, the classical case corresponding to the poset of positive integers ordered by divisibility. T. Leinster [Doc. Math. 13, 21–49 (2008; Zbl 1139.18009)] revisited Rota’s work, extending the notion of Euler characteristic to finite categories \(\boldsymbol{A}\) to get the notion of magnitude \(\chi(\boldsymbol{A})\).
Upon enumerating the objects of a finite category \(\boldsymbol{A}\) with \(n=\left\vert \mathrm{Ob}(\boldsymbol{A})\right\vert \), there is an algebra homomorphism between the incidence algebra \(R(\boldsymbol{A})\) and the matrices \(\mathrm{Mat}_{n\times n}(\mathbb{Q})\). Every matrix with complex coefficients has a unique Moore-Penrose pseudo-inverse, characterized by a system of equations (Theorem 3.1). In particular, the \(\zeta\) function is represented by a matrix \(Z\), whose pseudoinverse \(Z^{+}\) is the matricial representation of a pseudo-Möbius function. This paper establishes that one might use \(Z^{+}\) to check if a given finite category has magnitude and subsequently compute it. The magnitude arises as the sum of all the entries of the pseudoinverse matrix \(Z^{+}\) (Corollary 4.8). The main result (Theorem 4.1) gives a formula for the computation of the magnitude of a matrix in terms of its pseudoinverse and as such applies in a broader generality.