Putnam 1968 B5

Let $p$ be a prime number. Let $J$ be the set of all $2 \times 2$ matrices $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ whose entries are chosen from $\{0, 1, 2, \dots, p-1\}$ and satisfy the conditions $a+d \equiv 1 \pmod{p}$ and $ad-bc \equiv 0 \pmod{p}$. Determine how many members $J$ has.