In the image, you're asked to determine the determinant of matrix $B$, given that the determinant of matrix $A$ is 100. Matrix $B$ is obtained by multiplying the third row of matrix $A$ by 3, which means:

$B = A \quad \text{with} \quad R_3 \rightarrow 3R_3$

Now, let's go through the steps:

1. Rule for Row Multiplication and Determinants: When you multiply one row of a matrix by a scalar, the determinant of the matrix is multiplied by that scalar. In this case, you're multiplying the third row of $A$ by 3, so the determinant of $B$ will be:

   $\text{det}(B) = 3 \times \text{det}(A)$

2. Given that $\text{det}(A) = 100$, you can compute:

   $\text{det}(B) = 3 \times 100 = 300$

Therefore, the determinant of matrix $B$ is 300.

Would you like more details on any part of this calculation? Here are 5 follow-up questions to deepen your understanding:

1. Why does multiplying a row by a scalar affect the determinant this way?
2. What would happen to the determinant if two rows of a matrix were swapped?
3. How would the determinant change if multiple rows were scaled in a matrix?
4. Could you compute the determinant of $A$ manually? How is that done?
5. What properties of determinants make them useful in linear algebra?

Tip: Always remember that elementary row operations like swapping rows or multiplying rows by a scalar directly influence the determinant.