咨询：3×3矩阵对角元对应2×2行列式之和是否有专属名称

Great question! Let's unpack this step by step to clarify what your expression represents and whether it has a standard name.

First, let's restate your matrix and expression for clarity:
Given the 3×3 matrix:
$$ A = \begin{bmatrix} a & b & c\ d & e & f\ g & h & i \end{bmatrix} $$

You're investigating the sum:
$$ (ei-hf)+(ai-gc)+(ae-db) $$

Breaking Down Each Term #

Each component of your sum is a standard matrix concept called a minor:

• ei - hf: This is the minor of element (a) (written (M_{11})), calculated by taking the determinant of the 2×2 submatrix left after removing the first row and first column of (A).
• ai - gc: This is the minor of element (e) (written (M_{22})), the determinant of the submatrix left after removing the second row and second column.
• ae - db: This is the minor of element (i) (written (M_{33})), the determinant of the submatrix left after removing the third row and third column.

Does This Sum Have a Formal Name? #

There isn't a single universally recognized, shorthand name for this exact expression (unlike well-defined terms like "determinant" or "trace"). However, it's commonly described using precise terminology as:

• The sum of the main diagonal minors of the 3×3 matrix, or
• The sum of the principal 2×2 minors (since each minor comes from a principal submatrix—one that retains the same row and column indices as the original matrix).

In some specialized contexts, you might hear it referred to as a "trace-like minor sum" (drawing a parallel to the trace, which is the sum of the main diagonal elements), but this is not a formal, standard term.

内容的提问来源于stack exchange，提问作者Learn_and_Share