不使用初等变换法计算三阶矩阵的逆矩阵，可以借助行列式的性质，运用公式$A^{-1}=\frac{A_d}{|A|}$来计算，其中$A_d$是A的代数余子式矩阵，$|A|$是A的行列式的值。即$A^{-1}=\frac{\begin{vmatrix}a_{22}a_{33}-a_{23}a_{32}&-(a_{12}a_{33}-a_{13}a_{32})&a_{12}a_{23}-a_{13}a_{22}\\-(a_{21}a_{33}-a_{23}a_{31})&a_{11}a_{33}-a_{13}a_{31}&-(a_{11}a_{23}-a_{21}a_{13})\\a_{21}a_{32}-a_{31}a_{22}&-(a_{11}a_{32}-a_{31}a_{12})&a_{11}a_{22}-a_{21}a_{12}\end{vmatrix}}{\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}}$，经过计算即可得出三阶矩阵A的逆矩阵$A^{-1}$。