Linear and multilinear algebra; matrix theory Proposition

Adding a multiple of one row to another preserves the determinant

Statement

Let $K$ be a commutative ring, let $A$ be an $n \times n$ matrix over $K$ , and let $c \in K$ . Let $i \ne j$ , and let $B$ be the matrix that results when $c$ times row $j$ of $A$ is added to row $i$ . Then $\det B = \det A$ , where $\det$ is the determinant.

Remarks

By linearity in row $i$ , $\det B = \det A + c \det A'$ , where $A'$ has two equal rows; so $\det A' = 0$ . This is the reason why Gaussian elimination computes determinants efficiently. The same holds for columns.

Reading path

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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
Adding a multiple of one row to another preserves the determinant statement

Tag	`01ZTQBK64Q`
URI	`m2://01ZTQBK64Q`
Kind	statement, proposition
Writing	W0 stub
MSC	15A15
Status	This copy of the site has no build data. Run `m2 status 01ZTQBK64Q`.
Uses	Determinant

Adding a multiple of one row to another preserves the determinant

Statement

Remarks

Reading path

Further reading

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