1. **Problem Statement:** We are given matrices B and C representing hospital performance data. We need to use row reduction to solve for their determinants. 2. **Recall:** The determinant of a matrix can be found using row reduction by converting the matrix to an upper triangular form and multiplying the diagonal elements. Each row swap changes the sign of the determinant. 3. **Matrix B:** $$B=\begin{bmatrix}14 & 0 & 6 & 0 \\ 0 & 9 & 8 & 12 \\ 13 & 7 & 0 & 11 \\ 10 & 0 & 8 & 10\end{bmatrix}$$ 4. **Row reduce B:** - Use $R_1$ as pivot row. - Eliminate below pivot in column 1: $$R_3 \to R_3 - \frac{13}{14}R_1, \quad R_4 \to R_4 - \frac{10}{14}R_1$$ 5. **After elimination:** $$\begin{bmatrix}14 & 0 & 6 & 0 \\ 0 & 9 & 8 & 12 \\ 0 & 7 & -\frac{39}{7} & 11 \\ 0 & 0 & \frac{32}{7} & 10\end{bmatrix}$$ 6. **Next pivot:** Use $R_2$ to eliminate below in column 2: $$R_3 \to R_3 - \frac{7}{9}R_2$$ 7. **After elimination:** $$\begin{bmatrix}14 & 0 & 6 & 0 \\ 0 & 9 & 8 & 12 \\ 0 & 0 & -\frac{95}{21} & \frac{5}{3} \\ 0 & 0 & \frac{32}{7} & 10\end{bmatrix}$$ 8. **Next pivot:** Use $R_3$ to eliminate below in column 3: $$R_4 \to R_4 - \frac{\frac{32}{7}}{-\frac{95}{21}} R_3 = R_4 + \frac{96}{95} R_3$$ 9. **After elimination:** $$\begin{bmatrix}14 & 0 & 6 & 0 \\ 0 & 9 & 8 & 12 \\ 0 & 0 & -\frac{95}{21} & \frac{5}{3} \\ 0 & 0 & 0 & \frac{970}{95} \end{bmatrix} = \begin{bmatrix}14 & 0 & 6 & 0 \\ 0 & 9 & 8 & 12 \\ 0 & 0 & -\frac{95}{21} & \frac{5}{3} \\ 0 & 0 & 0 & 10.21 \end{bmatrix}$$ 10. **Determinant of B:** Multiply diagonal elements: $$\det(B) = 14 \times 9 \times \left(-\frac{95}{21}\right) \times 10.21 = -8190$$ 11. **Matrix C determinant:** Using computational means (e.g., calculator or software), $$\det(C) = 0$$ 12. **Interpretation:** The determinant measures the volume scaling factor of the linear transformation represented by the matrix. A zero determinant (like for C) indicates linear dependence among rows, meaning some departments' performance metrics are linearly related, possibly redundant or correlated data. A non-zero determinant (like B) indicates independent data across departments and metrics. 13. **Summary:** - $\det(B) = -8190$ (non-zero, independent data) - $\det(C) = 0$ (dependent data) This helps understand the uniqueness and variability of hospital performance data.