Question

If $A$ is a square matrix of order $3$ and if $\det (A)=3,$ then $\det [adj\{adj(adjA)\}]$ is equal to

• A. ${81}^2$
• B. $81$
• C. $729$
• D. $27$

Answer 1

Answer: (B)

Given, det $(A)=3$ and $n=3$
(order of A) Now, adj
$(adjA)=|A{|}^{n-2}A$
$={(3)}^{3-2}A$
$=3A$
$adj\{adj(adjA)\}=adj(3A)$
$=3^{n-1}adjA$
$=9adjA$ $|adj\{adj(adjA)\}=|9adjA|$
$=9|adjA|$
$=9|A{|}^{n-1}$
$=9{(3)}^{3-1}$
$=9\times 9=81$