Question

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

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

Answer 1

Answer: (B)

$81$

Answer 2

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