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: <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 ''n'' by ''n'' matrix is invertible exactly when its determinant is nonzero.
==Applications==
Quaternionic matrices are used in [[quantum mechanics]]<ref>{{cite journal |author= N. Rösch |title=Time-reversal symmetry, Kramers' degeneracy and the algebraic eigenvalue problem |year=1983 |journal=[[Chemical Physics]] |volume=80 |issue=1–2 |pages=1–5 |doi=10.1016/0301-0104(83)85163-5}}</ref> and in the treatment of [[multibody problem]]s.<ref>{{cite book |title=Quaternionic and Clifford calculus for physicists and engineers |author=Klaus Gürlebeck |author2=Wolfgang Sprössig |chapter=Quaternionic matrices |pages=32–34 |publisher=Wiley |year=1997 |isbn=978-0-471-96200-7}}</ref>
==References==
{{reflist}}
*{{cite book |title=Matrix groups for undergraduates|first=Kristopher|last=Tapp
|publisher=AMS Bookstore|year=2005|isbn=
|url=http://books.google.com/books?id=Un_15Im3NhUC&pg=PA11#v=onepage&q&f=false}}
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