Question

If A is a square matrix such that $A\left(AdjA\right)=\left[\begin{array}{ccc}4& 0& 0\\ 0& 4& 0\\ 0& 0& 4\end{array}\right]$, then $\frac{|Adj\left(AdjA\right)}{|AdjA|}$ is equal to

• A. $16$
• B. $46$
• C. $15$
• D. $12$

Answer 1

The correct option is A $16$

$A\left(AdjA\right)=\left[\begin{array}{ccc}4& 0& 0\\ 0& 4& 0\\ 0& 0& 4\end{array}\right]=4I$

$A\left(AdjA\right)=\left|A\right|I$

$\Rightarrow △=4$

$\frac{\left|Adj\right(AdjA\left)\right|}{\left|AdjA\right|}=\frac{{\left|A\right|}^{{(n-1)}^2}}{{\left|A\right|}^{n-1}}$

$={\left|A\right|}^{n-1}$

$={\left|A\right|}^{3-1}$

$={\left(4\right)}^2$

$=16$

Option A