Integrating a matrix Is it possible to integrate a matrix?
I've been working through a problem and come up with $$\int_{t_0}^t  \begin{pmatrix}\sin(s)\cdot\cos(\beta s)\\ \cos(s)\cdot\cos(\beta s)\end{pmatrix}ds$$
I'm integrating from $t_0$ to $t$
Can this be done or do we think I went wrong somewhere?
 A: Matrices form a vector space. Therefore, you can simply integrate them componentwise. 
In detail. Let $A:t\mapsto A(t)$ be a function from a real interval $I$ to the space of $m\times n$ real matrices. 
Every entry $a_{ij}$ is a real function of a real variable. If all entries are integrable functions, then you can define the integral of the matrix as the matrix of the integrals:
$$\int A(t)\,dt := \left( \int a_{ij}(t)\,dt \right).$$
For your problem:
$$\int  \begin{pmatrix}\sin(s)&\cos(\beta s)\\ \cos(s)&\cos(\beta s)\end{pmatrix}ds = 
\begin{pmatrix}\int\sin(s)\,ds &\int\cos(\beta s)\,ds\\ \int\cos(s)\,ds&\int\cos(\beta s)\,ds\end{pmatrix} = \begin{pmatrix}-\cos(s)&\frac{1}{\beta}\sin(\beta s)\\ \sin(s)&\frac{1}{\beta}\sin(\beta s)\end{pmatrix}. $$
There is a more sophisticated operation, in case the matrix in question belongs to a Lie algebra: ordered exponentiation. It is to integration as exponentiation is to multiplication, and permits to go from a Lie algebra element (intuitively, a differential transformation) to a group element (a whole transformation). 
In this case, you need a $n\times n$ matrix-valued function.
It is explained quite well here:
http://en.wikipedia.org/wiki/Ordered_exponential.
