To solve the problem, we first need to analyze the dimensions of matrices $A$ and $B$ and determine the order of the product matrix $AB$.

Given:

• $A$ is a $2 \times 3$ matrix (2 rows and 3 columns).
• $B$ is a $3 \times 4$ matrix (3 rows and 4 columns).

When multiplying two matrices $A_{m \times n}$ and $B_{p \times q}$, the multiplication is possible if and only if the number of columns in the first matrix ($n$) is equal to the number of rows in the second matrix ($p$). The resulting matrix $AB$ will have dimensions $m \times q$.

In this case:

• The number of columns of $A$ is 3, and the number of rows of $B$ is 3, so the multiplication is possible.
• The resulting matrix $AB$ will have dimensions $2 \times 4$.

Conclusion:
The order of matrix $AB$ is $\boxed{2 \times 4}$.

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. What conditions must be met for the multiplication of two matrices to be possible?
2. How can you determine the order of a resulting matrix after multiplication?
3. Can you provide an example where matrix multiplication is not possible?
4. How does the associative property apply to matrix multiplication?
5. What is the significance of matrix dimensions in real-world applications?

Tip: Always check the inner dimensions (columns of the first matrix and rows of the second) to determine if matrix multiplication is valid.