Answer
\begin{array}{l}{\operatorname{det}(A)=\left|\begin{array}{ccc}{\sqrt{\pi}} & {e^{2}} & {e^{-1}} \\ {\sqrt{67}} & {1 / 30} & {2001} \\ {\pi} & {\pi^{2}} & {\pi^{3}}\end{array}\right| \approx 9601.88}\end{array}
Work Step by Step
Given $$ A=\left[ \begin{array}{ccc}{\sqrt{\pi}} & {e^{2}} & {e^{-1}} \\ {\sqrt{67}} & {1 / 30} & {2001} \\ {\pi} & {\pi^{2}} & {\pi^{3}}\end{array} \right]$$
Since, if we have$$A=\left[ \begin{array}{lll}{a_{11}} & {a_{12}} & {a_{13}} \\ {a_{21}} & {a_{22}} & {a_{23}} \\ {a_{31}} & {a_{32}} & {a_{33}}\end{array} \right]$$
we get
\begin{aligned}\operatorname{det}(A)& =\left|\begin{array}{lll}{a_{11}} & {a_{12}} & {a_{13}} \\ {a_{21}} & {a_{22}} & {a_{23}} \\ {a_{31}} & {a_{32}} & {a_{33}}\end{array}\right|
&=a_{11}( a_{22} a_{33}-a_{23} a_{32})-a_{12}( a_{21} a_{33}-a_{23} a_{31})+a_{13}( a_{21} a_{32}-a_{22} a_{31})\\
& =a_{11}( a_{22} a_{33}-a_{23} a_{32})-a_{12}( a_{21} a_{33}-a_{23} a_{31})+a_{13}( a_{21} a_{32}-a_{22} a_{31})\\
&=a_{11} a_{22} a_{33}+a_{12} a_{23} a_{31}+a_{13} a_{21} a_{32}-a_{11} a_{23} a_{32}-a_{12} a_{21} a_{33}-a_{13} a_{22} a_{31}\\
\end{aligned}
Therefore, we get
\begin{aligned} \operatorname{det}(A)&=\left|\begin{array}{ccc}{\sqrt{\pi}} & {e^{2}} & {e^{-1}} \\ {\sqrt{67}} & {1 / 30} & {2001} \\ {\pi} & {\pi^{2}} & {\pi^{3}}\end{array}\right|\\
& =\sqrt{\pi}(1 / 30)\left(\pi^{3}\right)+e^{2}(2001)(\pi)+e^{-1}(\sqrt{67})\left(\pi^{2}\right)-e^{-1}(1 / 30)(\pi)- \sqrt{\pi}(2001)\left(\pi^{2}\right)-e^{2}(\sqrt{67})\left(\pi^{3}\right)\\
& \approx 9601.88
\end{aligned}
