The sizes of two matrices $A$ and $B$ are given. Find the sizes of the product $A B$ and the product $B A$, whenever these products exist. $A$ is $4 \times 5$, and $B$ is $5 \times 4$. Find the size of the product $\mathrm{AB}$. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The size of product $A B$ is B. The product $A B$ does not exist.

Step 1 ：The sizes of two matrices $A$ and $B$ are given. $A$ is $4 \times 5$, and $B$ is $5 \times 4$.

Step 2 ：The product of two matrices exists if the number of columns in the first matrix is equal to the number of rows in the second matrix.

Step 3 ：In this case, the number of columns in matrix A is 5, and the number of rows in matrix B is also 5, so the product AB exists.

Step 4 ：The size of the product matrix is determined by the number of rows in the first matrix and the number of columns in the second matrix.

Step 5 ：So, the size of the product AB is 4x4.

Step 6 ：Final Answer: The size of product $A B$ is \(\boxed{4 \times 4}\).