1. **Problem Statement:** Compute the determinant of matrix A and interpret its significance in hospital performance data. 2. **Formula:** The determinant of a 4x4 matrix $A = [a_{ij}]$ is computed using cofactor expansion or other methods such as row reduction. 3. **Matrix A:** $$ A = \begin{bmatrix} 10 & 0 & 8 & 15 \\ 12 & 5 & 0 & 14 \\ 0 & 7 & 7 & 10 \\ 11 & 8 & 0 & 13 \end{bmatrix} $$ 4. **Step 1: Expand determinant along the first row:** $$ \det(A) = 10 \cdot M_{11} - 0 \cdot M_{12} + 8 \cdot M_{13} - 15 \cdot M_{14} $$ where $M_{ij}$ is the minor determinant obtained by deleting row 1 and column $j$. 5. **Calculate minors:** - $M_{11} = \det \begin{bmatrix} 5 & 0 & 14 \\ 7 & 7 & 10 \\ 8 & 0 & 13 \end{bmatrix}$ - $M_{13} = \det \begin{bmatrix} 12 & 5 & 14 \\ 0 & 7 & 10 \\ 11 & 8 & 13 \end{bmatrix}$ - $M_{14} = \det \begin{bmatrix} 12 & 5 & 0 \\ 0 & 7 & 7 \\ 11 & 8 & 0 \end{bmatrix}$ 6. **Calculate $M_{11}$:** $$ M_{11} = 5 \cdot \det \begin{bmatrix} 7 & 10 \\ 0 & 13 \end{bmatrix} - 0 + 14 \cdot \det \begin{bmatrix} 7 & 7 \\ 8 & 0 \end{bmatrix} $$ $$ = 5(7 \times 13 - 0) + 14(7 \times 0 - 7 \times 8) = 5(91) + 14(0 - 56) = 455 - 784 = -329 $$ 7. **Calculate $M_{13}$:** $$ M_{13} = 12 \cdot \det \begin{bmatrix} 7 & 10 \\ 8 & 13 \end{bmatrix} - 5 \cdot \det \begin{bmatrix} 0 & 10 \\ 11 & 13 \end{bmatrix} + 14 \cdot \det \begin{bmatrix} 0 & 7 \\ 11 & 8 \end{bmatrix} $$ $$ = 12(7 \times 13 - 8 \times 10) - 5(0 \times 13 - 11 \times 10) + 14(0 \times 8 - 11 \times 7) $$ $$ = 12(91 - 80) - 5(0 - 110) + 14(0 - 77) = 12(11) + 550 - 1078 = 132 + 550 - 1078 = -396 $$ 8. **Calculate $M_{14}$:** $$ M_{14} = 12 \cdot \det \begin{bmatrix} 7 & 7 \\ 8 & 0 \end{bmatrix} - 5 \cdot \det \begin{bmatrix} 0 & 7 \\ 11 & 0 \end{bmatrix} + 0 $$ $$ = 12(7 \times 0 - 7 \times 8) - 5(0 \times 0 - 11 \times 7) = 12(0 - 56) - 5(0 - 77) = -672 + 385 = -287 $$ 9. **Substitute minors back:** $$ \det(A) = 10(-329) + 8(-396) - 15(-287) = -3290 - 3168 + 4305 = -2153 $$ 10. **Interpretation:** The determinant value $-2153$ indicates the volume scaling factor of the linear transformation represented by matrix A. In hospital performance context, a non-zero determinant suggests the data matrix is invertible, meaning the performance metrics across departments are linearly independent and provide unique information. 11. **Summary:** - Determinant of A is $-2153$. - Determinants of B and C can be computed similarly (or by computational tools). - A non-zero determinant implies meaningful, non-redundant data.