Answer
See explanation
Work Step by Step
The column-row expansions of $G_{k}$ and $G_{k+1}$ are:
$G_{k}=X_{k} \cdot X_{k}^{T}=$
$=\operatorname{col}_{1}\left(X_{k}\right) \cdot \operatorname{row}_{1}\left(X_{k}^{T}\right)+\cdots+\operatorname{col}_{k}\left(X_{k}\right) \cdot \operatorname{row}_{k}\left(X_{k}^{T}\right)$
$G_{k+1}=X_{k+1} \cdot X_{k+1}^{T}=$
$=\operatorname{col}_{1}\left(X_{k+1}\right) \cdot \operatorname{row}_{1}\left(X_{k+1}^{T}\right)+\cdots+\operatorname{col}_{k}\left(X_{k+1}\right) \cdot \operatorname{row}_{k}\left(X_{k+1}^{T}\right)+\operatorname{col}_{k+1}\left(X_{k+1}\right) \cdot \operatorname{row}_{k+1}\left(X_{k+1}^{T}\right)=$
$=\operatorname{col}_{1}\left(X_{k}\right) \cdot \operatorname{row}_{1}\left(X_{k}^{T}\right)+\cdots+\operatorname{col}_{k}\left(X_{k}\right) \cdot \operatorname{row}_{k}\left(X_{k}^{T}\right)+\operatorname{col}_{k+1}\left(X_{k+1}\right) \cdot \operatorname{row}_{k+1}\left(X_{k+1}^{T}\right)=$
$=G_{k}+\operatorname{col}_{k+1}\left(X_{k+1}\right) \cdot \operatorname{row}_{k+1}\left(X_{k+1}^{T}\right)$
Since you can see update from $G_{k}$ to $G_{k+1}$ can be made by adding matrix $\operatorname{col}_{k+1}\left(X_{k+1}\right) \cdot \operatorname{row}_{k+1}\left(X_{k+1}^{T}\right)$
