Given that
$\begin{bmatrix} 1&4 &3 \\2 &7 &9 \\5 &8 &a \end{bmatrix}=\begin{bmatrix}l_{11} &0 &0 \\ l_{21}&l_{22} &0 \\ l_{31}&l_{32} &-53 \end{bmatrix}.\begin{bmatrix} 1&u_{12} &u_{13} \\ 0& 1& u_{23}\\ 0&0 &1 \end{bmatrix}$
Now,
$\begin{bmatrix} 1&4 &3 \\2 &7 &9 \\5 &8 &a \end{bmatrix}=\begin{bmatrix}l_{11} &l_{11}u_{12} &l_{11}u_{13} \\ l_{21}&l_{21}u_{12}+l_{22} &l_{21}u_{13}+l_{22}u_{23} \\ l_{31}&l_{31}u_{12}+l_{32} &l_{31}u_{13}+l_{32}u_{23}-53 \end{bmatrix}$
Now we got $l_{11}=1,l_{21}=2,l_{31}=5$
$l_{11}u_{12}=4\Rightarrow u_{12}=4$
$l_{11}u_{13}=3\Rightarrow u_{13}=3$
$l_{21}u_{12}+l_{22}=7\Rightarrow l_{22}=-1$
$l_{31}u_{12}+l_{32}=8\Rightarrow l_{32}=-12$
$l_{21}u_{13}+l_{22}u_{23}=9\Rightarrow u_{23}=-3$
$l_{31}u_{13}+l_{32}u_{23}-53=a$
$\Rightarrow a=5\times 3+(-12)\times (-3)-53$
$\Rightarrow a=15+36-53$
$\Rightarrow a=51-53$
$\Rightarrow a=-2$