Question

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

• A. Is less than 256
• B. Has 21 divisors
• C. Greater than 256
• D. Is an odd number

Answer 1

The correct option is A Is less than 256

Let ABCD = E
$\therefore |adj\left(adj\right(adj\left(adj\right(ABCD\left)\right)\left)\right)|$
$=|adj\left(adj\right(adj\left(adj\right(E\left)\right)\left)\right)|$
$=|E{|}^{(3-1{)}^4}=|E{|}^{16}$
$=|ABCD{|}^{16}=|A{|}^{16}|B{|}^{16}|C{|}^{16}|D{|}^{16}$
$=|A{|}^{16}.(|A{|}^{3-1}{)}^{16}.(|A{|}^{(3-1{)}^2}{)}^{16}.(|A{|}^{(3-1{)}^3}{)}^{16}$
$=|A{|}^{16+32+64+128}=A^K$
$\Rightarrow K=240$; which is less than 256.