The electric potential in a region of space is given by V(x, y, z)...

Electric potential is defined as work done per unit charge, $V = \dfrac{W}{q}$.

The dimensional formula of work $W$ is $[M L^2 T^{-2}]$ and that of charge $q$ is $[I T]$.

The dimensional formula of electric potential $V$ is $\dfrac{[M L^2 T^{-2}]}{[I T]} = [M L^2 T^{-3} I^{-1}]$.

Given the relation $V(x, y, z) = A(xy + yz + zx)$, the term $(xy + yz + zx)$ represents the product of two lengths, so its dimensional formula is $[L^2]$.

Using the principle of dimensional homogeneity, the dimensional formula of $A$ is:

$[A] = \dfrac{[V]}{[L^2]} = \dfrac{[M L^2 T^{-3} I^{-1}]}{[L^2]} = [M T^{-3} I^{-1}]$