Let's solve this problem step by step. It involves understanding how row operations affect the determinant of a matrix.

(a) Row Operations on Matrix $A$

The operations applied are:

1. $R_1$ is swapped with $R_2$: Swapping two rows changes the sign of the determinant.
2. $R_3$ is replaced by $\frac{1}{3} R_3$: Scaling a row by $\frac{1}{3}$ scales the determinant by $\frac{1}{3}$.
3. $R_4$ is replaced by $R_4 + R_1$: Adding a multiple of one row to another does not change the determinant.

Let $\det(A)$ be the determinant of matrix $A$. After these operations: $\det(B) = -\frac{1}{3} \det(A)$ Here, the negative sign comes from the row swap, and the scaling by $\frac{1}{3}$ comes from the scaling of $R_3$.

Answer for (a): $\det(B) = -\frac{1}{3} \det(A)$.

(b) Row Operations on Matrix $C$

Given the row operations and the matrix $D$:

1. $R_1$ is swapped with $R_2$: Changes the sign of the determinant.
2. $R_2$ is replaced by $3 R_2$: Scaling a row by 3 scales the determinant by 3.
3. $R_3$ is replaced by $R_3 + 5 R_2$: Adding a multiple of one row to another does not change the determinant.

Given that: $D = \begin{bmatrix} -1 & -7 & 0 \\ 0 & 6 & -1 \\ 0 & 0 & 3 \end{bmatrix}$ we know that $\det(D) = \det(C) \cdot (-1) \cdot 3$, where:

• $-1$ comes from the swap,
• $3$ comes from scaling $R_2$.

Calculate $\det(D)$: $\det(D) = (-1)(6)(3) = -18$ Thus: $\det(C) = \frac{\det(D)}{-3} = \frac{-18}{-3} = 6$

Answer for (b): $\det(C) = 6$.

Summary

• (a) $\det(B) = -\frac{1}{3} \det(A)$
• (b) $\det(C) = 6$

Would you like further clarification or step-by-step breakdowns of any specific part? Here are 5 related questions to deepen understanding:

1. How do elementary row operations affect the determinant of a matrix?
2. What is the determinant of a matrix after multiple row swaps?
3. Can scaling multiple rows simultaneously affect the determinant similarly?
4. How is the determinant impacted if you replace a row by adding it to a scaled version of another row?
5. Can you provide examples of matrices with determinant zero and explain why?

Tip: When performing row operations, always remember how each operation impacts the determinant, especially swaps and scalings!