a time (LiLjLk), then four at a time, and so forth, until there are no more. Compute the determinant Δ and cofactors Δk. Apply the formula. Examples Circuit containing two-port The transfer function from Vin to V2 is desired. There is only one forward path: Vin to V1 to I2 to V2 with gain There are three loops: V1 to I1 to V1 with gain V2 to I2 to V2 with gain V1 to I2 to V2 to I1 to V1 with gain note: L1 and L2 do not touch each other whereas L3 touches both of the other loops. note: the forward path touches all the loops so all that is left is 1. Digital IIR biquad filter Digital filters are often diagramed as signal flow graphs. There are two loops Note, the two loops touch so there is no term for their product. There are three forward paths All the forward paths touch all the loops so Servo The signal flow graph has six loops. They are: There is one forward path: The forward path touches all the loops therefore the co-factor And the gain from input to output is Equivalent matrix form Mason's rule can be stated in a simple matrix form. Assume is the transient matrix of the graph where is the sum transmittance of branches from node m toward node n. Then, the gain from node m to node n of the graph is equal to , where , and is the identity matrix. Mason's Rule is also particularly useful for deriving the z-domain transfer function of discrete networks that have inner feedback loops embedded within outer feedback loops (nested loops). If the discrete network can be drawn as a signal flow graph, then the application of Mason's Rule will give that network's z-domain H(z) transfer function. Complexity and computational applications Mason's Rule can grow factorially, because the enumeration of paths in a directed graph grows dramatically. To see this consider the complete directed graph on vertices, having an edge between every pair of vertices. There is a path form to for each of the permutations of the intermediate vertices. Thus Gaussian elimination is more efficient in the general case. Yet Mason's rule characterizes the transfer functions of interconnected systems in a way which is simultaneously algebraic and combinatorial, allowing for general statements and other computations in algebraic systems theory. While numerous inverses occur during