A 3 . 1 Laplace Expansion of Determinant and Permanent
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چکیده
A3.1 Laplace Expansion of Determinant and Permanent Definition A3.1.1. Let [n] = {1, . . . , n}, I = {i1, . . . , ik} ⊆ [n] and J = {j1, . . . , jk} ⊆ [n]. If A is an n× n matrix, define the k × k matrix A(I; J) = (ãrs) of A to be the matrix with entries ãrs = airjs . Also define the (n− k)× (n− k) complementary matrix A(I; J), where I and J are the complements of I and J respectively. Instead of A({i}, {j}) we shall simply write A(i, j). Exercise A3.1.2. (Determinant expansion by a row) Suppose A is an n × n matrix and fix i ∈ [n]. Show that the following formulas hold.
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