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The determinant of a matrix *M* is written as either EQN:|\b{M}| or det( *M* ).

* The determinant of a 2x2 matrix is the /area/ scale-factor.

* The determinant of a 3x3 matrix is the /volume/ scale-factor.

The determinant of a 2x2 matrix EQN:\left[\begin{matrix}a&b\\c&d\end{matrix}\right] is EQN:\left|\begin{matrix}a&b\\c&d\end{matrix}\right|=ad-bc

Determinants can be used to find vector products:

** *a* x *b* = EQN:\left|\begin{matrix}\b{i}&\b{j}&\b{k}\\a_1&a_2&a_3\\b_1&b_2&b_3\end{matrix}\right|=\b{i}(a_2b_3-a_3b_2)-\b{j}(a_1b_3-a_3b_1)+\b{k}(a_1b_2-a_2b_1)

where *a* is the vector EQN:\left[\begin{matrix}a_1\\a_2\\a_3\end{matrix}\right] , *b* is the vector EQN:\left[\begin{matrix}b_1\\b_2\\b_3\end{matrix}\right] and *i* , *j* and *k* are unit vectors.

Determinants can be used to find the vector cross product.

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* See also Matrices