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The image presents a question involving two matrices \( A \) and \( B \), with a transformation applied to matrix \( A \) to get matrix \( B \).

The problem states:
- \( A = \begin{pmatrix} -5 & -1 & 4 & -3 \\ 4 & -2 & 0 & 2 \\ 2 & 1 & 3 & 0 \\ -5 & -4 & 3 & -1 \end{pmatrix} \)
- \( B \) is obtained by multiplying the third row of \( A \) by 3, denoted as \( 3R_3 \).
- The determinant of matrix \( A \) is given as 100.
- The question asks to find the determinant of matrix \( B \).

### Solution:
1. The determinant of a matrix changes when a row operation is performed. Specifically, when one row of a matrix is multiplied by a scalar, the determinant is also multiplied by that scalar.

2. In this case, matrix \( B \) is obtained from matrix \( A \) by multiplying the third row of \( A \) by 3. Therefore, the determinant of \( B \) can be calculated by multiplying the determinant of \( A \) by 3.

3. Given:
   \[
   \det(A) = 100
   \]
   and since the third row is multiplied by 3, the new determinant is:
   \[
   \det(B) = 3 \times \det(A) = 3 \times 100 = 300
   \]

Thus, the determinant of matrix \( B \) is **300**.

Would you like further clarification or more details on any part of the process?

### Here are 5 related questions to explore:
1. What happens to the determinant if two rows of a matrix are swapped?
2. How do elementary row operations affect the determinant of a matrix?
3. What are the conditions under which a matrix has a zero determinant?
4. How can we calculate the determinant of a 4x4 matrix using cofactor expansion?
5. What is the geometric interpretation of a matrix's determinant?

### Tip:
When performing elementary row operations, remember that only scaling operations affect the determinant multiplicatively. Row swaps change the sign, and adding multiples of one row to another leaves the determinant unchanged.

The determinant of matrix A is 100. Matrix B is obtained by multiplying the third row of A by 3. What is the determinant of B?

Learn how to find the determinant of a matrix B obtained by multiplying a row of matrix A by a scalar. Given the determinant of A as 100, this solution explores how to calculate det(B) using the multiplication rule for row operations in linear algebra.

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