Example of finding the inverse of a 3-by-3 matrix by using Cramer's rule

To obtain column $j$ of $A^{-1}$, solve $Ax=e_j$ using Cramer's rule: $x_i=\det(A_i(e_j))/\det A$. For $A=\operatorname{diag}(2,3,4)$, this gives columns $(1/2,0,0)^T$, $(0,1/3,0)^T$, $(0,0,1/4)^T$, hence $A^{-1}=\operatorname{diag}(1/2,1/3,1/4)$. Determinant-based inversion is mainly theoretical, not an efficient large-system method.Copy