View source for Equivalent matrices

''$A$ and $B$ over a ring $R$''

Matrices such that $A$ can be transformed into $B$ by a sequence of elementary row-and-column transformations, that is, transformations of the following three types: a) permutation of the rows (or columns); b) addition to one row (or column) of another row (or column) multiplied by an element of $R$; or c) multiplication of a row (or column) by an invertible element of $R$.  Equivalently, $B$ is obtained from $A$ by multiplication on left or right by a sequence of matrices each of which is either a) a [[permutation matrix]]; b) an [[elementary matrix]]; c) an invertible [[diagonal matrix]].

Equivalence in this sense is an [[equivalence relation]].

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