Jeffrey M. Dambacher, Hiram W. Li, and Philippe A. Rossignol. 2002. Relevance of community structure in assessing indeterminacy of ecological predictions. Ecology 83:1372-1385.

The determinant (det) of a second order system is: .

Calculation of the determinant of larger systems is by expansion of its matrix minors (min) along either its columns or its rows. For a third order system, the A_1,1 minor is formed by deletion of the first row and column, giving

.

Calculation of the determinant for the entire matrix thus becomes:

. .

Expansion of the determinant can proceed along any row or column, provided the correct sign is applied in the terms of the minors, according to the formula: -1^i+j. Determinants of matrices greater than third order are calculated by expansion with the minor method, but calculations become tedious.

Stability is equated with self-damping or negative overall feedback in a system (Levins 1975). Overall feedback is defined as the determinant of a system. The concept of self-damping being synonymous with negative overall feedback, however, is confounded by determinants of stable even-sized systems always being positive. A sign convention is therefore employed of (-1ⁿ⁺¹) det A, which ensures that stability can be equated with negative overall feedback in both even- and odd-sized systems.

Another source of potential confusion is associated with the denominator of Eq. 9, where -A^-1 = adjoint -A / det -A. In stable systems with negative overall feedback (following the convention of the -1ⁿ⁺¹ multiplier), the sign of the det -A term in the denominator will always be positive in both even- and odd-sized systems (the -1ⁿ⁺¹ multiplier is not applied to the det -A term in Eq. 9). Thus the det -A term will not alter the equality of the signs of corresponding elements of the inverse and adjoint matrices. In unstable systems with positive overall feedback, however, the det -A term will be negative in both even- and odd-sized systems. Thus the sign of the adjoint -A_ij elements will be opposite to those of corresponding -A^-1_ij elements. Confusion notwithstanding, this inconsistency leaves us with a useful condition for unstable systems, for the adjoint -A_ij elements will always have response signs as if a system were stable. As a consequence, systems that are conditionally or neutrally stable can be assessed in terms of an expected or possible equilibrium behavior—a result not possible through use of the inverse matrix, which depends on a nonzero determinant for matrix inversion.

Calculation of a matrix permanent (per) is similar to the determinant, but without subtraction within matrix minors, and with an all positive sign convention for expansion terms: (+1)^i+j (Marcus and Minc 1964, Minc 1978, Eves 1980). Thus the permanent of a second order matrix becomes

and that of a third order

. .

Cofactors of a matrix (C_ij) are determinants of each matrix minor, with the same (-1)^i+j sign convention applied in the expansion, such that the matrix of all cofactors (C) becomes

. .

The adjoint matrix is simply a transposed matrix of cofactors, such that adjoint A_ij = C_ji. Calculation of the absolute feedback matrix (T) from Eq. 10 is similar to the above cofactor calculations (transposed), but it uses the matrix permanent instead of the determinant, and all expansion terms are positive.