Editing Invertible matrix (section)

=== Invertible matrix theorem ===
Let {{math|'''A'''}} be a square {{mvar|n}}-by-{{mvar|n}} matrix over a [[field (mathematics)|field]] {{mvar|K}} (e.g., the field {{tmath|\mathbb R}} of real numbers). The following statements are equivalent, i.e., they are either all true or all false for any given matrix:<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Invertible Matrix Theorem|url=https://mathworld.wolfram.com/InvertibleMatrixTheorem.html|access-date=2020-09-08|website=mathworld.wolfram.com|language=en}}</ref>
* {{math|'''A'''}} is invertible, i.e. it has an inverse under matrix multiplication, i.e., there exists a {{math|'''B'''}} such that {{math|1='''AB''' = '''I'''{{sub|''n''}} = '''BA'''}}. (In that statement, "invertible" can equivalently be replaced with "left-invertible" or "right-invertible" in which one-sided inverses are considered.)
* The linear transformation mapping {{math|'''x'''}} to {{math|'''Ax'''}} is invertible, i.e., it has an inverse under function composition. (There, again, "invertible" can equivalently be replaced with either "left-invertible" or "right-invertible".)
* The [[transpose]] {{math|'''A'''<sup>T</sup>}} is an invertible matrix.
* {{math|'''A'''}} is [[Row equivalence|row-equivalent]] to the {{mvar|n}}-by-{{mvar|n}} [[identity matrix]] {{math|'''I'''{{sub|''n''}}}}.
* {{math|'''A'''}} is [[Row equivalence|column-equivalent]] to the {{mvar|n}}-by-{{mvar|n}} identity matrix {{math|'''I'''{{sub|''n''}}}}.
* {{math|'''A'''}} has {{mvar|n}} [[pivot position]]s.
* {{math|'''A'''}} has full [[Rank (linear algebra)|rank]]: {{math|1=rank '''A''' = ''n''}}.
* {{math|'''A'''}} has a trivial [[Kernel (linear algebra)|kernel]]: {{math|1=ker('''A''') = {'''0'''}.}}
* The linear transformation mapping {{math|'''x'''}} to {{math|'''Ax'''}} is bijective; that is, the equation {{math|1='''Ax''' = '''b'''}} has exactly one solution for each {{math|'''b'''}} in {{mvar|K{{sup|n}}}}. (There, "bijective" can equivalently be replaced with "[[injective]]" or "[[surjective]]".)
* The columns of {{math|'''A'''}} form a [[basis of a vector space|basis]] of {{mvar|K{{sup|n}}}}. (In this statement, "basis" can equivalently be replaced with either "linearly independent set" or "spanning set")
* The rows of {{math|'''A'''}} form a basis of {{mvar|K{{sup|n}}}}. (Similarly, here, "basis" can equivalently be replaced with either "linearly independent set" or "spanning set")
* The [[determinant]] of {{math|'''A'''}} is nonzero: {{math|det '''A''' ≠ 0}}. In general, a square matrix over a [[commutative ring]] is invertible if and only if its determinant is a [[Unit (ring theory)|unit]] (i.e. multiplicatively invertible element) of that ring.
* The number 0 is not an [[eigenvalue]] of {{math|'''A'''}}. (More generally, a number <math>\lambda</math> is an eigenvalue of {{math|'''A'''}} if the matrix <math>\mathbf{A}-\lambda \mathbf{I}</math> is singular, where {{math|'''I'''}} is the identity matrix.)
* The matrix {{math|'''A'''}} can be expressed as a finite product of [[Elementary matrix|elementary matrices]].