In the LU decomposition of the matrix $\begin{bmatrix} 2 & 2 \\ 4 & 9 \end{bmatrix}$, if the diagonal elements of $U$ are both 1, then the lower diagonal entry $l_{22}$ of $L$ is _______.

Let matrix $A = \begin{bmatrix} 2 & 2 \\ 4 & 9 \end{bmatrix}$.

Given that the diagonal elements of $U$ are both 1, the matrices $L$ and $U$ take the form:

Multiplying $L$ and $U$:

Equating entries with $A = \begin{bmatrix} 2 & 2 \\ 4 & 9 \end{bmatrix}$:

1. $l_{11} = 2$

2. $l_{11}u_{12} = 2 \implies 2u_{12} = 2 \implies u_{12} = 1$

3. $l_{21} = 4$

4. $l_{21}u_{12} + l_{22} = 9 \implies (4)(1) + l_{22} = 9 \implies l_{22} = 9 - 4 = 5$

Thus, the lower diagonal entry $l_{22}$ is 5 [1].