Verify directly that the three functions \(E_1\), \(E_2\), \(E_3\) defined by (5-6), (5-7), and (5-8) are identical.
Solution
Goal: Verify directly by algebraic expansion that the three cofactor formulas \(E_1, E_2, E_3\) for the \(3 \times 3\) matrix \(A = (A_{ij})\) from Hoffman and Kunze are identical.
1. Definition and expansion of \(E_1(A)\) (Laplace expansion along row 1, formula 5-6):Proof:
1.1. Formula (5-6) defines\[E_1(A) = A_{11}(A_{22} A_{33} - A_{23} A_{32}) - A_{12}(A_{21} A_{33} - A_{23} A_{31}) + A_{13}(A_{21} A_{32} - A_{22} A_{31}).\]
1.2. Distributing the terms:\[E_1(A) = A_{11} A_{22} A_{33} - A_{11} A_{23} A_{32} - A_{12} A_{21} A_{33} + A_{12} A_{23} A_{31} + A_{13} A_{21} A_{32} - A_{13} A_{22} A_{31}.\]
2. Definition and expansion of \(E_2(A)\) (Laplace expansion along row 2, formula 5-7):Proof:
2.1. Formula (5-7) defines\[E_2(A) = -A_{21}(A_{12} A_{33} - A_{13} A_{32}) + A_{22}(A_{11} A_{33} - A_{13} A_{31}) - A_{23}(A_{11} A_{32} - A_{12} A_{31}).\]
2.2. Distributing the terms:\[E_2(A) = -A_{21} A_{12} A_{33} + A_{21} A_{13} A_{32} + A_{22} A_{11} A_{33} - A_{22} A_{13} A_{31} - A_{23} A_{11} A_{32} + A_{23} A_{12} A_{31}.\]
2.3. Rearranging into standard factor order:\[E_2(A) = A_{11} A_{22} A_{33} - A_{11} A_{23} A_{32} - A_{12} A_{21} A_{33} + A_{12} A_{23} A_{31} + A_{13} A_{21} A_{32} - A_{13} A_{22} A_{31}.\]
3. Definition and expansion of \(E_3(A)\) (Laplace expansion along row 3, formula 5-8):Proof:
3.1. Formula (5-8) defines\[E_3(A) = A_{31}(A_{12} A_{23} - A_{13} A_{22}) - A_{32}(A_{11} A_{23} - A_{13} A_{21}) + A_{33}(A_{11} A_{22} - A_{12} A_{21}).\]
3.2. Distributing the terms:\[E_3(A) = A_{31} A_{12} A_{23} - A_{31} A_{13} A_{22} - A_{32} A_{11} A_{23} + A_{32} A_{13} A_{21} + A_{33} A_{11} A_{22} - A_{33} A_{12} A_{21}.\]
3.3. Rearranging into standard factor order:\[E_3(A) = A_{11} A_{22} A_{33} - A_{11} A_{23} A_{32} - A_{12} A_{21} A_{33} + A_{12} A_{23} A_{31} + A_{13} A_{21} A_{32} - A_{13} A_{22} A_{31}.\]
4. Conclusion:Proof:All three expanded expressions consist of the exact same 6 signed monomials:\[E_1(A) = E_2(A) = E_3(A) = \sum_{\sigma \in S_3} \operatorname{sgn}(\sigma) A_{1\sigma(1)} A_{2\sigma(2)} A_{3\sigma(3)}.\]