Alphabeta MathHow statement and proof provenance work
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The determinant is alternating and multilinear in the rows as well as in the columns
Statement
For , determinant on over a commutative ring is multilinear and alternating in the rows. Interchanging two rows negates the determinant.
Facts & Assumptions
Given: A square matrix whose selected row is varied, or whose two selected rows are interchanged.
Transpose leaves determinant unchanged (For every square matrix over a commutative ring, ).
Determinant is alternating and multilinear in columns (The Leibniz determinant is column-multilinear, alternating and normalized over every commutative ring).
Transpose is defined by , so it interchanges rows and columns (Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose).
Proof
Transposition sends a selected row, a row sum, a row scaling or a row swap to the corresponding operation on a selected column, without changing determinant.
Apply the column multilinearity and alternation of [L2] to the transpose, then apply [L1] again. The resulting identities are exactly row multilinearity and row alternation for the original matrix.