1. **Problem statement:** Calculate the determinant of matrix $A = (a_{ij})_{4 \times 4}$ where $$a_{ij} = \begin{cases} \log_2(32 - i), & i < j \\ \log_3\left(\frac{27}{81}\right), & i > j \\ \sec(2\pi) + \sqrt[4]{81}, & i = j \end{cases}$$ 2. **Recall the formulas and rules:** - The determinant of a $4 \times 4$ matrix can be calculated using the Laplace expansion theorem. - Important values: - $\log_2(32 - i)$ for $i=1,2,3,4$. - $\log_3\left(\frac{27}{81}\right) = \log_3(3^{-1}) = -1$. - $\sec(2\pi) = 1$ since $\cos(2\pi) = 1$. - $\sqrt[4]{81} = 3$ because $3^4 = 81$. 3. **Calculate diagonal elements $a_{ii}$:** $$a_{ii} = \sec(2\pi) + \sqrt[4]{81} = 1 + 3 = 4$$ 4. **Calculate elements above diagonal $a_{ij}$ for $ij$:** All equal to $-1$ as shown above. 6. **Construct matrix $A$: ** $$A = \begin{pmatrix} 4 & \log_2(31) & \log_2(31) & \log_2(31) \\ -1 & 4 & \log_2(30) & \log_2(30) \\ -1 & -1 & 4 & \log_2(29) \\ -1 & -1 & -1 & 4 \end{pmatrix}$$ (Note: The problem states $a_{ij} = \log_2(32 - i)$ for $i