$\text{claim:} $The game will end when all cells will be empty $\text{proof:} $Assume for the sake of the contradiction that the cell with cordinates $(a,b)$ (a-th row,b-th column)is not empty but game ended.Then in a row $a$ and in a column $b$ we must have at least one empty cells in each .(Otherwise the game will continue).Assume $(a,d);(c,b)$ are this cells. Then let $r_{i};c_{i}$ denotes number of usings of row $i$ and column $i$ during the gamerespectively.Then $r_{a}+c_{b}<99;r_{a}+c_{d}=99;r_{c}+c_{b}=99 \rightarrow r_{c}+c_{b} +r_{a}+c_{d}=2 * 99 \rightarrow r_{c}+c_{d}>99$. A contradiction $\text{answer:}$ So in anyway we must have $99n$ turns which implies iff $n$ is odd then $A$ will win $\blacksquare$