[Math] What does it mean to multiply a real matrix by a complex scalar

Question

In this answer https://math.stackexchange.com/a/219508/27609 it is noted, that multiplying a matrix $A$ by a scalar $s$ is the same as multiplying a matrix $A$ by a diagonal matrix ${\rm diag}(c,c,\ldots, c)$ of the appropriate size. But is this also the case, if $s$ is a complex number? What does it mean when one multiplies a matrix by e.g. the imaginary unit $i$. Does it simply mean $$i *\pmatrix{a&b\\c&d} = \pmatrix{i&0\\0&i} \pmatrix{a&b\\c&d}?$$ Or does it mean $$i *\pmatrix{a&b\\c&d} = \pmatrix{0&1\\-1&0} \pmatrix{a&b\\c&d}?$$ The latter could also make sense, because the imaginary unit $i$ is equivalent to the 2×2 matrix $J=\pmatrix{0&1\\-1&0},$ since $J^2=\pmatrix{-1&0\\0&-1}$ (see e.g. Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$), but this would obviously give a different result when multiplying with a matrix.

Answer 1

Each matrix element is multiplied by the scalar, not matter if it is real-valued or complex.