\text{A matrix is given as:}\left[\begin{array}{cc}9& 15\\ 15& 50\end{array}\right].\text{ By performing Cholesky decomposition,}{l}_{22}\text{ of the lower triangular matrix is}\text{ \_\_\_\_\_\_ }\text{(in integer).}

The correct answer is 5.

To find the elements of the lower triangular matrix obtained via Cholesky decomposition, we write the symmetric matrix as the product of a lower triangular matrix and its transpose. Let the given matrix be $A$:
$$A=\left[\begin{array}{cc}9& 15\\ 15& 50\end{array}\right]$$

We want to find a lower triangular matrix $L$:
$$L=\left[\begin{array}{cc}{l}_{11}& 0\\ l_{21}& l_{22}\end{array}\right]$$
such that:
$$A=L{L}^T$$

Multiplying matrix $L$ by its transpose $L^T$, we obtain:
$$L{L}^T=\left[\begin{array}{cc}{l}_{11}& 0\\ l_{21}& l_{22}\end{array}\right]\left[\begin{array}{cc}{l}_{11}& l_{21}\\ 0& l_{22}\end{array}\right]=\left[\begin{array}{cc}{l}_{11}^2& l_{11}{l}_{21}\\ l_{11}{l}_{21}& l_{21}^2+l_{22}^2\end{array}\right]$$

By equating the elements of the resulting matrix product to the elements of matrix $A$, we can solve for each component of $L$ step-by-step:

1. Find $l_{11}$ by equating the first diagonal element:
$$l_{11}^2=9$$ Since the diagonal elements of a Cholesky factor must be positive, we have:
$$l_{11}=\sqrt{9}=3$$

2. Find $l_{21}$ by equating the off-diagonal element:
$$l_{11}{l}_{21}=15$$ Substituting $l_{11}=3$ into the equation:
$$3\cdot l_{21}=15$$
$$l_{21}=\frac{15}{3}=5$$

3. Find $l_{22}$ by equating the second diagonal element:
$$l_{21}^2+l_{22}^2=50$$ Substituting $l_{21}=5$ into the equation:
$$5^2+l_{22}^2=50$$
$$25+l_{22}^2=50$$
Subtracting 25 from both sides yields:
$$l_{22}^2=25$$ Since the diagonal elements of $L$ must be positive, we find:
$$l_{22}=\sqrt{25}=5$$

Therefore, the entry $l_{22}$ of the lower triangular matrix is 5.