$\Rightarrow A$ adj $(2 A )=-4 I$ $......(I)$

Now, $E =\left| A ^{4}\right|+\mid A ^{10}-(\operatorname{adj}(2 A ))^{10}$

$=(-2)^{4}+\frac{\left| A ^{20}- A ^{10}(\operatorname{adj} 2 A )^{10}\right|}{| A |^{10}}$

$=16+\frac{\left| A ^{20}-(\operatorname{Aadj}(2 A ))^{10}\right|}{| A |^{10}}$

$=16+\frac{\left| A ^{20}-2^{10} I \right|}{2^{10}}$ (from

Now, characteristic roots of $A$ are $2$ and $-1 .$

So, characteristic roots of $A ^{20}$ are $2^{10}$ and $1$. Hence, $\left( A ^{20}-2^{10} I \right)\left( A ^{20}- I \right)=0$

$\Rightarrow\left| A ^{20}-2^{10} I \right|=0\left(\right.$ as $\left. A ^{20} \neq I \right)$

$\Rightarrow E =16$ Ans.