linear algebra - Question about elementary row operations with block matrices

Given two $n \times n$ matrices $A$ and $B$, form a new block matrix

$$P := \begin{bmatrix}I_n&B\\-A&0\end{bmatrix}$$

Then by using only elementary row operations, show that $P$ can be transformed into

$$P' := \begin{bmatrix}I_n&B\\0&AB\end{bmatrix}$$

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The solution to this problem is:

$$P = \begin{bmatrix}I_n&B\\-A&0\end{bmatrix} \sim \begin{bmatrix}I_n&B\\-A + AI_n &0 + AB\end{bmatrix} \sim \begin{bmatrix}I_n&B\\0&AB\end{bmatrix}$$

I don't understand this solution. Why can $A$ be multiplied from the left on the first half of the matrix and then be added to the second half of the matrix to form a sequence of elementary row operations?