The most natural way to define eM is to imitate the power series for ex :
In general,
Evaluate
eA =6pt
etA =6pt
eA+B =6pt
eAeB =6pt
DetCD-1 =6pt
eDtCD-1 =6pt
Note that one of the properties of exponents that holds for real numbers fails for matrices. The equality eA+B = eAeB requires that AB = BA, and this property fails to hold for the matrices in the example. However, exponentiation preserves the property of similarity, as demonstrated by DetCD-1 = eDtCD-1.