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===The invertible matrix theorem===
Let '''A''' be a square ''n'' by ''n'' matrix over a [[field (mathematics)|field]] ''K'' (for example the field '''R''' of real numbers). The following statements are equivalent:
: '''A''' is invertible, i.e. '''A''' has an inverse, is nonsingular, or is
: '''A''' is [[Row equivalence|row-equivalent]] to the ''n''-by-''n'' [[identity matrix]] '''I'''<sub>''n''</sub>.
: '''A''' is [[Row equivalence|column-equivalent]] to the ''n''-by-''n'' [[identity matrix]] '''I'''<sub>''n''</sub>.
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