Given the matrix equation , where and is a matrix, I want to show that is the solution where and is the identity matrix.

I start out by writing the above series in summation notation which gives me . I can now take a time derivative of the sum to give me . Since the first term of the series after differentiation is , I can rewrite and reduce the sum to give me . Now, I can pull out a single matrix term to give me . I can now simplify the sum once again to give me . This is now equivalent to . Assuming that is the solution, I can differentiate it with respect to time to give me . I just showed that , which I can plug in to give me the equation . Since , I am left with my original matrix equation

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