Linear and multilinear algebra; matrix theory Proposition

Determinant is multilinear in the rows

Statement

Let $K$ be a commutative ring, let $n \ge 1$ , and let $1 \le i \le n$ . Let $A$ , $B$ , $C$ be $n \times n$ matrices over $K$ that agree outside row $i$ . Let $a$ , $b$ , $c$ be row $i$ of $A$ , $B$ , $C$ , and assume $c = sa + tb$ with $s, t \in K$ . Then $\det C = s \det A + t \det B,$ where $\det$ is the determinant.

Remarks

This is property 1 of the definition of the determinant. The determinant is not additive in the whole matrix: in general $\det(A + B) \ne \det A + \det B$ . By $\det A^T = \det A$ , the determinant is also multilinear in the columns.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Identity matrix object
Matrix definition
Vector space definition
Square matrix definition
Linear map definition
Multilinear map definition
Alternating multilinear map definition
Determinant definition
Determinant is multilinear in the rows statement

Tag	`0PPG793642`
URI	`m2://0PPG793642`
Kind	statement, proposition
Writing	W0 stub
MSC	15A15
Status	This copy of the site has no build data. Run `m2 status 0PPG793642`.
Uses	Determinant

Determinant is multilinear in the rows

Statement

Remarks

Reading path

Further reading

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