,

$$\begin{eqnarray*} A^{-1} & = & \frac{1}{\det A}\operatorname{adj}\left(A\right)\\ \operatorname{adj}\left(A\right) & = & \left(\det A\right)A^{-1}\\ \det\left(\operatorname{adj}\left(A\right)\right) & = & \det\left[\left(\det A\right)A^{-1}\right] \end{eqnarray*}$$

ใช้กฏการดึงค่าคงตัวออกจาก $\det$ ได้

$$\begin{eqnarray*} \det\left(\operatorname{adj}\left(A\right)\right) & = & \left(\det\left(A\right)\right)^{3}\det\left(A^{-1}\right)\\ & = & \left(\det\left(A\right)\right)^{3}\frac{1}{\det A}\\ & = & \left(\det\left(A\right)\right)^{2} \end{eqnarray*}$$

,

$$\begin{eqnarray*} \det\left(A^{2}\right) & = & \det\left(3A\right)\\ \left(\det\left(A\right)\right)^{2} & = & 3^{3}\det A\\ \frac{\left(\det\left(A\right)\right)^{2}}{\det A} & = & 27\\ \det\left(A\right) & = & 27 \end{eqnarray*}$$