$AA$

$\underset{–}{\times AA}$

$AA$

$\underset{–}{+AA\times }$

$\underset{–}{ABA}$

What is the value of A ?

The correct option is B $1$
Now, we are given an operation at the unit's place, i.e., $A\times A=A$. If we go by the options and put $A$ value option (1) $A=2$, then,

$2\times 2=4$, i.e., $A\times A\ne A$. Thus this is incorrect.
Similarly, in case of options (2) & (3)

$4\times 4\ne 4$ and $7\times 7\ne 7$ respectively, therefore, both these options are also discarded.
Thus option (4) $1\times 1=1$ is the correct answer.