Let \(K\) be a commutative ring with identity. Define a function \(D\) on \(3 \times 3\) matrices over \(K\) by Laplace cofactor expansion along the first row: \[ D(A) = A_{11} \det \begin{bmatrix} A_{22} & A_{23} \\ A_{32} & A_{33} \end{bmatrix} - A_{12} \det \begin{bmatrix} A_{21} & A_{23} \\ A_{31} & A_{33} \end{bmatrix} + A_{13} \det \begin{bmatrix} A_{21} & A_{22} \\ A_{31} & A_{32} \end{bmatrix}. \] Show that \(D\) is alternating and 3-linear as a function of the columns of \(A\).
Solution
Goal: Prove that the Laplace expansion \(D(c_1, c_2, c_3)\) on \(3 \times 3\) matrices is multilinear (3-linear) and alternating in its column vectors.
1. Setting and Column Representation:Proof:
1.1. Let the columns of \(A \in M_3(K)\) be \(c_1 = \begin{bmatrix} A_{11} \\ A_{21} \\ A_{31} \end{bmatrix}\), \(c_2 = \begin{bmatrix} A_{12} \\ A_{22} \\ A_{32} \end{bmatrix}\), \(c_3 = \begin{bmatrix} A_{13} \\ A_{23} \\ A_{33} \end{bmatrix}\).
1.2. The formula for \(D(A) = D(c_1, c_2, c_3)\) expands as:\[D(c_1, c_2, c_3) = A_{11}(A_{22} A_{33} - A_{32} A_{23}) - A_{12}(A_{21} A_{33} - A_{31} A_{23}) + A_{13}(A_{21} A_{32} - A_{31} A_{22}).\]
1.3. Expanding the terms fully:\[D(A) = A_{11} A_{22} A_{33} - A_{11} A_{32} A_{23} - A_{12} A_{21} A_{33} + A_{12} A_{31} A_{23} + A_{13} A_{21} A_{32} - A_{13} A_{31} A_{22}.\]
1.4. In permutation notation:\[D(A) = \sum_{\sigma \in S_3} \operatorname{sgn}(\sigma) A_{\sigma(1), 1} A_{\sigma(2), 2} A_{\sigma(3), 3} = \sum_{\tau \in S_3} \operatorname{sgn}(\tau) A_{1, \tau(1)} A_{2, \tau(2)} A_{3, \tau(3)}.\]
2. Proof of 3-Linearity:Proof:
2.1. In the fully expanded sum, each monomial contains exactly one entry from column 1 (\(A_{i1}\)), exactly one entry from column 2 (\(A_{j2}\)), and exactly one entry from column 3 (\(A_{k3}\)).
2.2. If column \(c_1\) is replaced by \(\alpha u + \beta v\), then for each row \(i\), \(A_{i1} = \alpha u_i + \beta v_i\).
2.3. By distributivity of multiplication over addition in \(K\):\[D(\alpha u + \beta v, c_2, c_3) = \alpha D(u, c_2, c_3) + \beta D(v, c_2, c_3).\]
2.4. By identical reasoning on columns 2 and 3, \(D\) is linear in \(c_2\) and linear in \(c_3\).
2.5. Therefore, \(D\) is a 3-linear function of the columns.
3. Proof that \(D\) is Alternating:Proof:
3.1. A multilinear function is alternating if \(D(A) = 0\) whenever two adjacent columns are equal.
3.2. Case 1: \(c_1 = c_2\).- Then \(A_{i1} = A_{i2}\) for all \(i \in \{1, 2, 3\}\).- In the definition of \(D(A)\):- The first two terms are \(A_{11}(A_{21} A_{33} - A_{31} A_{23}) - A_{11}(A_{21} A_{33} - A_{31} A_{23}) = 0\).- The third minor is \(\det \begin{bmatrix} A_{21} & A_{21} \\ A_{31} & A_{31} \end{bmatrix} = A_{21} A_{31} - A_{31} A_{21} = 0\).- Thus \(D(c_1, c_1, c_3) = 0 - 0 + A_{13}(0) = 0\).
3.3. Case 2: \(c_2 = c_3\).- Then \(A_{i2} = A_{i3}\) for all \(i \in \{1, 2, 3\}\).- The first minor is \(\det \begin{bmatrix} A_{22} & A_{22} \\ A_{32} & A_{32} \end{bmatrix} = 0\).- The second and third terms are:\[-A_{12}(A_{21} A_{32} - A_{31} A_{22}) + A_{12}(A_{21} A_{32} - A_{31} A_{22}) = 0.\]- Thus \(D(c_1, c_2, c_2) = A_{11}(0) - 0 = 0\).
3.4. Case 3: \(c_1 = c_3\).- By swapping columns, \(D(c_1, c_2, c_1) = 0\).
3.5. Since \(D(A)\) vanishes whenever any two columns coincide, \(D\) is alternating.
4. Conclusion:\(D\) is linear in each column entry and vanishes when any two columns are equal, so \(D\) is an alternating 3-linear function. Q.E.D.