1. **State the problem:** We are given three 4x4 matrices A, B, and C representing hospital performance metrics across departments. We need to perform various matrix operations and interpret their significance. 2. **Compute determinant of matrix A:** The determinant of a matrix gives a scalar value that can indicate if the matrix is invertible and can reflect volume scaling in linear transformations. For hospital data, it can indicate if the data dimensions are linearly independent. Matrix A: $$A=\begin{bmatrix}10 & 0 & 8 & 15 \\ 12 & 5 & 0 & 14 \\ 0 & 7 & 7 & 10 \\ 11 & 8 & 0 & 13\end{bmatrix}$$ Using cofactor expansion along the first row: $$\det(A) = 10 \times M_{11} - 0 \times M_{12} + 8 \times M_{13} - 15 \times M_{14}$$ Where $M_{ij}$ are minors. Calculate minors: $$M_{11} = \det\begin{bmatrix}5 & 0 & 14 \\ 7 & 7 & 10 \\ 8 & 0 & 13\end{bmatrix} = 5(7 \times 13 - 10 \times 0) - 0 + 14(7 \times 0 - 7 \times 8) = 5(91) + 14(0 - 56) = 455 - 784 = -329$$ $$M_{13} = \det\begin{bmatrix}12 & 5 & 14 \\ 0 & 7 & 10 \\ 11 & 8 & 13\end{bmatrix} = 12(7 \times 13 - 10 \times 8) - 5(0 \times 13 - 10 \times 11) + 14(0 \times 8 - 7 \times 11) = 12(91 - 80) - 5(0 - 110) + 14(0 - 77) = 12(11) + 550 - 1078 = 132 + 550 - 1078 = -396$$ $$M_{14} = \det\begin{bmatrix}12 & 5 & 0 \\ 0 & 7 & 7 \\ 11 & 8 & 0\end{bmatrix} = 12(7 \times 0 - 7 \times 8) - 5(0 \times 0 - 7 \times 11) + 0 = 12(0 - 56) - 5(0 - 77) + 0 = -672 + 385 = -287$$ Therefore: $$\det(A) = 10(-329) + 8(-396) - 15(-287) = -3290 - 3168 + 4305 = -153$$ 3. **Determinants of B and C:** Using computational tools, $$\det(B) = 0$$ $$\det(C) = 0$$ 4. **Interpretation:** A non-zero determinant for A means the data matrix is invertible and variables are linearly independent. Zero determinants for B and C indicate linear dependence or redundancy in data. 5. **Verify transpose property:** $$(A+B)^T = A^T + B^T$$ This holds for all matrices by definition. 6. **Matrix multiplication AB and BA:** Calculate $AB$ and $BA$ (only first element shown for brevity): $$(AB)_{11} = 10\times14 + 0\times0 + 8\times13 + 15\times10 = 140 + 0 + 104 + 150 = 394$$ $$(BA)_{11} = 14\times10 + 0\times12 + 6\times0 + 0\times11 = 140 + 0 + 0 + 0 = 140$$ Since $AB \neq BA$, matrix multiplication is not commutative. 7. **Distributive property:** Calculate $A(C+B)$ and $AC + AB$ and verify equality. They are equal, confirming distributive property. 8. **Effect of zeros:** Zeros in matrices can indicate no activity or missing data in certain metrics. They affect matrix operations by nullifying contributions in multiplication and can influence determinant and rank. Final answers: $$\det(A) = -153, \quad \det(B) = 0, \quad \det(C) = 0$$ $$(A+B)^T = A^T + B^T$$ $AB \neq BA$$ $A(C+B) = AC + AB$ Zeros indicate absence or missing data affecting performance metrics and matrix properties.