In this video, I calculate the exponential of a matrix and other fun matrix functions, like the cosine or the sine of a matrix, and as an application, I solve systems of differential equations like x’ = Ax or x” + A^2 x = 0. This video illustrates how powerful power series actually are: they allow us to define matrix functions, which is usually a hard thing to do. And, of course, some diagonalization comes into play as well. Enjoy this beautiful mix of calculus, linear algebra, and differential equations