Editing Matrix exponential (section)

=== The determinant of the matrix exponential ===
By [[Jacobi's formula]], for any complex square matrix the following [[trace identity]] holds:<ref>{{harvnb|Hall|2015}} Theorem 2.12</ref>
{{Equation box 1
|indent =
|equation = <math>\det\left(e^A\right) = e^{\operatorname{tr}(A)}~.</math>
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|border colour = #0073CF
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}}

In addition to providing a computational tool, this formula demonstrates that a matrix exponential is always an [[invertible matrix]]. This follows from the fact that the right hand side of the above equation is always non-zero, and so {{math|det(''e<sup>A</sup>'') ≠ 0}}, which implies that {{math|''e<sup>A</sup>''}} must be invertible.

In the real-valued case, the formula also exhibits the map
<math display="block">\exp \colon M_n(\R) \to \mathrm{GL}(n, \R)</math>
to not be [[surjective function|surjective]], in contrast to the complex case mentioned earlier. This follows from the fact that, for real-valued matrices, the right-hand side of the formula is always positive, while there exist invertible matrices with a negative determinant.