Question

Let $A$ be a matrix of order $3\times 3$ and matrices $B,C$ and $D$ are related such that $B=adj(A)C=adj(adjA),D=(adj(adj(adjA))).$ If $|adj(adj(adj(adjABCD)))|$ is $|A{|}^k$ , 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
$B=\mathrm{adj}(A)\cdot C=\mathrm{adj}(\mathrm{adj}(A))$
$D=\mathrm{adj}(\mathrm{adj}(\mathrm{adj}(A)))$
Let $ABCD=E$, then
$\mid \mathrm{adj}(adjadj(adjE)))i=\mid E$
$=\mid E$
$|ABCD|16=|A|B|C|D$
$=\mid A{\left(A^2\right)}^{16}(\mid A{)}^{16}(\mid A{)}^{16}$
$=1A=1A$
$\Rightarrow |A=|A$
$\Rightarrow K=240<256$