Matrices Question 4
Question 4 - 25 January - Shift 1
Let $x, y, z>1$ and
$ A= \begin{bmatrix} 1 & \log _{x} y & \log _{x} z \\ \log _{y} x & 2 & \log _{y} z \\ \log _{z} x & \log _{z} y & 3 \end{bmatrix} $
Then $|adj(adj A^{2})|$ is equal to
(1) $6^{4}$
(2) $2^{8}$
(3) $4^{8}$
(4) $2^{4}$
Show Answer
Answer: (2)
Solution:
Formula: Properties of Adjoint of a Matrix, Properties of Matrix Multiplication
$|A|=\frac{1}{\log x \cdot \log y \cdot \log z} \begin{vmatrix} \log x & \log y & \log z \\ \log x & 2 \log y & \log z \\ \log x & \log y & 3 \log z\end{vmatrix} =2$
$\Rightarrow|adj(adj A^{2})|=|A^{2}|^{4}=2^{8}$
