A dash of symmetry ......

Let $S$ be the set of all $3$ x $3$ matrices which have only $0$ 's and $1$ 's as entries. (The number of entries that can be $0$ can be any integer from $0$ to $9$ inclusive, as is the case for the number of entries that can be $1$ .)

The probability that a matrix, chosen at random from $S$ , is symmetric is $\dfrac{a}{b}$ , where $a$ and $b$ are positive coprime integers. Find $a + b$ .

(This post was inspired by this question .)