Let \(K\) be a subfield of the complex numbers and \(n\) a positive integer. Let \(j_1, \ldots, j_n\) and \(k_1, \ldots, k_n\) be positive integers not exceeding \(n\). For an \(n \times n\) matrix \(A\) over \(K\) define \[ D(A) = A(j_1, k_1)A(j_2, k_2) \cdots A(j_n, k_n). \] Prove that \(D\) is \(n\)-linear if and only if the integers \(j_1, \ldots, j_n\) are distinct.
Solution
1. Definition of \(n\)-linearity of row functions:
1.1. A function \(D: M_{n \times n}(K) \to K\) is \(n\)-linear as a function of the rows \(\alpha_1, \ldots, \alpha_n\) of \(A\) if for each row index \(r \in \{1, \ldots, n\}\), the mapping \(\alpha_r \mapsto D(\alpha_1, \ldots, \alpha_r, \ldots, \alpha_n)\) is linear when the remaining rows \(\alpha_i\) (\(i \neq r\)) are held fixed.In particular, scaling row \(\alpha_r\) by \(c \in K\) satisfies \(D(\alpha_1, \ldots, c\alpha_r, \ldots, \alpha_n) = c D(\alpha_1, \ldots, \alpha_r, \ldots, \alpha_n)\).
2. Direction (\(\Leftarrow\)): If \(j_1, \ldots, j_n\) are distinct, then \(D\) is \(n\)-linear:
2.1. If \(j_1, \ldots, j_n\) are distinct integers in \(\{1, \ldots, n\}\), then \(\{j_1, \ldots, j_n\} = \{1, \ldots, n\}\), so \(j\) is a permutation of \(\{1, \ldots, n\}\).
2.2. For any fixed row index \(r \in \{1, \ldots, n\}\), there is a unique index \(m \in \{1, \ldots, n\}\) such that \(j_m = r\).We can factor \(D(A)\) as:\[ D(A) = A(r, k_m) \prod_{i \neq m} A(j_i, k_i). \]
2.3. The term \(A(r, k_m)\) is a linear functional on the row vector \(\alpha_r = (A(r, 1), \ldots, A(r, n))\), while the product \(\prod_{i \neq m} A(j_i, k_i)\) depends only on the rows \(\alpha_i\) with \(i \neq r\) and is independent of \(\alpha_r\).Therefore \(D\) is linear in \(\alpha_r\).Since this holds for every \(r \in \{1, \ldots, n\}\), \(D\) is \(n\)-linear.
3. Direction (\(\Rightarrow\)): If \(D\) is \(n\)-linear, then \(j_1, \ldots, j_n\) are distinct:
3.1. Suppose for contradiction that \(j_1, \ldots, j_n\) are not distinct.By the Pigeonhole Principle, some integer \(r \in \{1, \ldots, n\}\) appears \(d \ge 2\) times in the sequence \((j_1, \ldots, j_n)\).
3.2. Let \(E\) be the \(n \times n\) matrix with all entries equal to \(1\).Then \(D(E) = 1^n = 1 \neq 0\).
3.3. For any \(c \in K\), let \(E(c)\) be the matrix obtained by multiplying row \(r\) of \(E\) by \(c\), while leaving all other rows unchanged.Because row \(r\) appears \(d\) times in the product \(D(A)\), each of these \(d\) factors is multiplied by \(c\), yielding:\[ D(E(c)) = c^d D(E) = c^d. \]
3.4. On the other hand, since \(D\) is \(n\)-linear, scaling row \(r\) by \(c\) must satisfy:\[ D(E(c)) = c D(E) = c. \]Therefore \(c^d = c\) for all \(c \in K\).Because \(K\) is a subfield of \(\mathbb{C}\), choosing \(c = 2 \in K\) gives \(2^d = 2\), which implies \(d = 1\), contradicting \(d \ge 2\).
4. Conclusion:\(D\) is \(n\)-linear if and only if \(j_1, \ldots, j_n\) are distinct. Q.E.D.