There’s a little more to it than that. Matrices have several properties like being symmetric or unitary (maybe even diagonal) that scalars don’t. Those enable rewritings to make systems more stable or computationally efficient.
That bears no relation to the symbolic differentiation in the OP though.
Even plain old multiplication and division, and even addition and subtraction have stability and efficiency problems on floats, which don't appear in symbolic solvers.
Even plain old multiplication and division, and even addition and subtraction have stability and efficiency problems on floats, which don't appear in symbolic solvers.