question on anticommutator

Let \psi(x) be a fermionic field. and \psi_bar = complex conjugate of
\psi * gamma_0 matrix

Now, we have the anticommutatior \psi_bar(y).\psi(x) + \psi(x).
\psi_bar(y) = propagator(y-x)

Now, the propagator is a matrix,
In the anticommutator, the first term is a scalar
(\psi_bar = a row_matrix*gamma_matrix = row_matrix; hence first term =
row * column matrix = scalar)
The second term is column_matrix * row_matrix = a matrix

So, I have a scalar + matrix = matrix
Not sure what this means? Is there an Identity matrix hidden in the
first term?

In applying the wick's theorem, is one replacing the scalars with
matrices?

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