Question

Let $A$ be a $3\times 3$ invertible matrix. If $\mid$ adj $(24A)\mid =\mathrm{adj}(3\mathrm{adj}(2A))\mid$, then $|A{|}^2$ is equal to :

• A. $6^6$
• B. $2^{12}$
• C. $2^6$
• D. $1$

Answer 1

Answer: (C)

$|\mathrm{adj}(24A)|=|\mathrm{adj}3(\mathrm{adj}2A)|$
$\Rightarrow |24a{|}^2=(3\mathrm{adj}(2A){)}^2$
$\Rightarrow {\left(24^3|A|\right)}^2={\left(3^3|\mathrm{adj}(2A)|\right)}^2$
$=3^6{\left(|2A{|}^2\right)}^2$
$\Rightarrow 24^6|A{|}^2={\left(24^3|A|\right)}^2=3^6\times 2^{12}|A{|}^4$
$\Rightarrow |A{|}^2=\frac{24^6}{3^6\times 2^{12}}=64$