Answer
\begin{aligned} \operatorname{det}(A)&=\left|\begin{array}{ccc}{2} & {3} & {-1} \\ {1} & {4} & {1} \\ {3} & {1} & {6}\end{array}\right|
=48
\end{aligned}
Work Step by Step
Given $$ A=\left[ \begin{array}{ccc}{2} & {3} & {-1} \\ {1} & {4} & {1} \\ {3} & {1} & {6}\end{array} \right]$$
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}{2} & {3} & {-1} \\ {1} & {4} & {1} \\ {3} & {1} & {6}\end{array}\right|\\
& =2(4)(6)+(3)(1)(3)+(-1)(1)(1)-(-1)(4)(3)-2(1)(1)-3(1)(6)\\
&=48
\end{aligned}