Question

Let $A$ be a matrix of order $3\times 3$ and matrices $B,C$ and $D$ are related such that $B=adj\left(\right.A\left.\right)C=adj\left(\right.adjA\left.\right),D=\left(\right.adj\left(\right.adj\left(\right.adjA\left.\right)\left.\right)\left.\right).$ If $\left|\right.adj\left(\right.adj\left(\right.adj\left(\right.adjABCD\left.\right)\left.\right)\left.\right)\left|\right.$ is $\left|\right.A\left|\right.^{k}$ , then $k$

• A. is less than $256$ .
• B. has $21$ divisors
• C. Cannot say
• D. is an odd number

Answer 1

Answer: (A)

Here, $A$ is a $3 \times 3$ matrix matrices $B, C$ and $D$ are related such that
$B=\operatorname{adj}(A) \cdot C=\operatorname{adj}(\operatorname{adj}(A))$
$D=\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(A)))$
Let $A B C D=E$, then
$\mid \operatorname{adj} \quad(a d j \quad a d j \quad(a d j \quad E))) i =\mid E$
$=\mid E$
$|A B C D| 16=|A| B|C| D$
$=\mid A\left(A^2\right)^{16}(\mid A)^{16}(\mid A)^{16}$
$=1 A=1 A$
$\Rightarrow \quad|A=| A$
$\Rightarrow K=240<256$