When we have functions of matricies, eg. \exp(kA), where k is a scalar and A is a matrix, how is that envisioned? Does the function operate on each element of the matrix individually?
No, the function does not act on each element of the matrix separately. You have to expand the function in a power series so that you can write the function of a matrix as a sum of terms that involve powers of that matrix. A matrix M to the n-th power is the consecutive matrix multiplication of n copies of the matrix M. See Boas Eqn. 6.17 for the example of \exp(kM) where k is a scalar.
\exp \left ( \left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ] \right ) \neq \left [ \begin{array}{cc} \exp(a) & \exp(b) \\ \exp(c) & \exp(d) \end{array} \right ]
\left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ]^3 = \left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ] \left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ] \left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ] \neq \left [ \begin{array}{cc} a^3 & b^3 \\ c^3 & d^3 \end{array} \right ]
\exp(kA) = 1 + kA + \frac{k^2A^2}{2} + \frac{k^3A^3}{3!} + \cdot , where it is understood that A^3 is the matrix multiplication of A A A.