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The number of purely real elements in a lower triangular representation of the given $3\times 3$ matrix, obtained through the given decomposition is ______________.

$$\begin{bmatrix} 2 &3 &3 \\ 3 &2 &1 \\ 3&1 & 7 \end{bmatrix}=\begin{bmatrix} a_{11} &0 &0 \\ a_{12}& a_{22} &0 \\ a_{13}&a_{23} &a_{33} \end{bmatrix}\begin{bmatrix} a_{11} &0 & 0\\ a_{12} &a_{22} &0 \\ a_{13}&a_{23} &a_{33} \end{bmatrix}^{\top}$$

  1. $5$
  2. $6$
  3. $8$
  4. $9$
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9 will be the answer as number of pure real elements in a lower triangular matrix.
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