Question

Let $A=\left[a_{ij}\right]$ be a $4\times 4$ matrix. If $a_{ij}=\{\begin{array}{c}2,\text{when}i=j\\ 0,\text{when}i\ne j\end{array}$,
then the value of $\left\{\frac{\text{det(adj(adjA))}}{7}\right\}$ is
(where $\{.\}$ represents the fractional part function)

• A. $\frac{1}{7}$
• B. $\frac{2}{7}$
• C. $\frac{3}{7}$
• D. $\frac{4}{7}$

Answer 1

The correct option is A $\frac{1}{7}$

$A=\left[\begin{array}{cccc}2& 0& 0& 0\\ 0& 2& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 2\end{array}\right]$

$\Rightarrow |A|=2^4$
We know that for a $n\times n$ matrix,
$|\text{adj}A|=|A{|}^{n-1}$
$\Rightarrow |\text{adj}A|=|A{|}^3=2^{12}$
$\Rightarrow |\text{adj(adj A)}|=|\text{adj}A{|}^3=2^{36}$

Now, $\left\{\frac{\text{det(adj(adj A))}}{7}\right\}$
$=\left\{\frac{2^{36}}{7}\right\}$
$=\left\{\frac{(7+1{)}^{12}}{7}\right\}=\frac{1}{7}$