Let $$S = \left\{\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} : a_{ij} \in \{0, 1, 2\}, a_{11} = a_{22}\right\}$$. Then the number of non-singular matrices in the set S is :

The set $$S$$ consists of all $$2 \times 2$$ matrices $$\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$$ where each entry $$a_{ij}$$ is in $$\{0, 1, 2\}$$ and $$a_{11} = a_{22}$$. Let $$a = a_{11} = a_{22}$$, $$b = a_{12}$$, and $$c = a_{21}$$. Then any matrix in $$S$$ has the form $$\begin{pmatrix} a & b \\ c & a \end{pmatrix}$$, with $$a, b, c \in \{0, 1, 2\}$$.

A matrix is non-singular if its determinant is non-zero. The determinant of $$\begin{pmatrix} a & b \\ c & a \end{pmatrix}$$ is $$a \cdot a - b \cdot c = a^2 - bc$$. So, we require $$a^2 - bc \neq 0$$.

Since $$a$$, $$b$$, and $$c$$ each have 3 possible values (0, 1, or 2), the total number of matrices in $$S$$ is $$3 \times 3 \times 3 = 27$$. We need to subtract the number of singular matrices (where $$a^2 - bc = 0$$) from 27 to find the non-singular ones.

We consider each possible value of $$a$$ separately:

Case 1: $$a = 0$$
Determinant is $$0^2 - bc = -bc$$. This is zero when $$bc = 0$$.
Total matrices: $$a = 0$$, $$b \in \{0,1,2\}$$, $$c \in \{0,1,2\}$$ → 9 matrices.
Singular when $$bc = 0$$:
- If $$b = 0$$, any $$c$$ (3 cases: $$c=0,1,2$$)
- If $$c = 0$$, any $$b$$ (but $$b=0$$ already counted) → add $$b=1, c=0$$ and $$b=2, c=0$$ → 2 more
Total singular: $$3 + 2 = 5$$.
Non-singular: $$9 - 5 = 4$$.
Alternatively, non-singular when $$b \neq 0$$ and $$c \neq 0$$, so $$b \in \{1,2\}$$, $$c \in \{1,2\}$$ → $$2 \times 2 = 4$$.

Case 2: $$a = 1$$
Determinant is $$1^2 - bc = 1 - bc$$. This is zero when $$bc = 1$$.
Total matrices: 9.
Singular when $$bc = 1$$:
- Only $$(b,c) = (1,1)$$ works (since $$1 \cdot 1 = 1$$)
- Other pairs: $$(0,0)=0$$, $$(0,1)=0$$, $$(0,2)=0$$, $$(1,0)=0$$, $$(1,2)=2$$, $$(2,0)=0$$, $$(2,1)=2$$, $$(2,2)=4$$ → none equal 1
So only 1 singular matrix.
Non-singular: $$9 - 1 = 8$$.

Case 3: $$a = 2$$
Determinant is $$2^2 - bc = 4 - bc$$. This is zero when $$bc = 4$$.
Total matrices: 9.
Singular when $$bc = 4$$:
- Only $$(b,c) = (2,2)$$ works (since $$2 \cdot 2 = 4$$)
- Other pairs give less than 4
So only 1 singular matrix.
Non-singular: $$9 - 1 = 8$$.

Summing non-singular matrices: $$4$$ (for $$a=0$$) + $$8$$ (for $$a=1$$) + $$8$$ (for $$a=2$$) = $$20$$.

Hence, the correct answer is Option D.