Matrices & Determinants
Adjoint of adjoint, determinant
nta_pyq_2023_jan
Grade 12
Question:
Let $x, y, z > 1$ and $A = \begin{pmatrix} 1 & \log_x y & \log_x z \\ \log_y x & 2 & \log_y z \\ \log_z x & \log_z y & 3 \end{pmatrix}$. Then $|\text{adj}(\text{adj}A^2)|$ is equal to
Step-by-Step Solution
Key Concept: Compute $|A|$ first using logarithm properties, then use $|\text{adj}(\text{adj}M)| = |M|^{(n-1)^2}$ for $n \times n$ matrix
$|A| = \frac{1}{\log x \cdot \log y \cdot \log z} \begin{vmatrix} \log x & \log y & \log z \\ \log x & 2\log y & \log z \\ \log x & \log y & 3\log z \end{vmatrix} = 2$. So $|A^2| = 4$. $|\text{adj}(\text{adj}A^2)| = |A^2|^{(3-1)^2} = 4^4 = 2^8$. Answer: (2)
Correct Answer: $2^8$