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Again, since multiplication is not commutative some care must be taken in the order of the factors.<ref>{{cite book |title=Matrix groups for undergraduates|first=Kristopher|last=Tapp
|publisher=AMS Bookstore|year=2005|isbn=0-8218-3785-0 |pages=11 ''ff''
|url=https://books.google.com/books?id=Un_15Im3NhUC&pg=PA11
==Determinants==
There is no natural way to define a [[determinant]] for (square) quaternionic matrices so that the values of the determinant are quaternions.<ref>{{cite journal |author=Helmer Aslaksen |title=Quaternionic determinants |year=1996 |journal=[[The Mathematical Intelligencer]] |volume=18 |number=3 |pages=57–65 |doi=10.1007/BF03024312|s2cid=13958298 }}</ref> Complex valued determinants can be defined however.<ref>{{cite journal |author=E. Study |title=Zur Theorie der linearen Gleichungen |year=1920 |journal=[[Acta Mathematica]] |volume=42 |number=1 |pages=1–61 |language=German |doi=10.1007/BF02404401|doi-access=free }}</ref> The quaternion ''a'' + ''bi'' + ''cj'' + ''dk'' can be represented as the 2×2 complex matrix
: <math>\begin{bmatrix}~~a+bi & c+di \\ -c+di & a-bi \end{bmatrix}.</math>
This defines a map Ψ<sub>''mn''</sub> from the ''m'' by ''n'' quaternionic matrices to the 2''m'' by 2''n'' complex matrices by replacing each entry in the quaternionic matrix by its 2 by 2 complex representation. The complex valued determinant of a square quaternionic matrix ''A'' is then defined as det(Ψ(''A'')). Many of the usual laws for determinants hold; in particular, an [[square matrix|''n'' by ''n'' matrix]] is invertible if and only if its determinant is nonzero.
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