If  a diagonal matrix A  of order $3\times 3$ is commutative with every square matrix of order $3\times 3$ under multiplication and $tr\left(A\right)=12$ then the value of ${\left|A\right|}^{1/2}$ is-----------

$\text{ let }A=\left[\begin{array}{ccc}{\mathrm{d}}_1& 0& 0\\ 0& {\mathrm{d}}_2& 0\\ 0& 0& {\mathrm{d}}_3\end{array}\right]$

A is commutative with every square matrix of order $3\times 3$ under multiplication

$\Rightarrow {\mathrm{d}}_1={\mathrm{d}}_2={\mathrm{d}}_3$

$tr\left(A\right)=12\Rightarrow d_1=d_2=d_3=4$

$\left|A\right|=d_1{d}_2{d}_3=4^3=64$

Every diagonal element is 4

$\left|A\right|=64$

$|\mathrm{A}{|}^{\frac{1}{2}}=8$