The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system

$\mathrm{A}\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has exactly two distinct solutions, is

Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices :

$$ \mathrm{T}_{\mathrm{p}}=\left\{\mathrm{A}=\left[\begin{array}{ll} a & b \\ c & a \end{array}\right]: a, b, c \in\{0,1,2, \ldots, p-1\}\right\} $$

Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices :

$$ \mathrm{T}_{\mathrm{p}}=\left\{\mathrm{A}=\left[\begin{array}{ll} a & b \\ c & a \end{array}\right]: a, b, c \in\{0,1,2, \ldots, p-1\}\right\} $$

The number of A in $\mathrm{T}_p$ such that the trace of A is not divisible by $p$ but $\operatorname{det}(\mathrm{A})$ is divisible by $p$ is

[Note : The trace of a matrix is the sum of its diagonal entries.]

Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices :

$$ \mathrm{T}_{\mathrm{p}}=\left\{\mathrm{A}=\left[\begin{array}{ll} a & b \\ c & a \end{array}\right]: a, b, c \in\{0,1,2, \ldots, p-1\}\right\} $$