<P> More generally, if A is "near" the invertible matrix X in the sense that </P> <Dl> <Dd> lim n → ∞ (I − X − 1 A) n = 0 o r lim n → ∞ (I − A X − 1) n = 0 (\ displaystyle \ lim _ (n \ to \ infty) (\ mathbf (I) - \ mathbf (X) ^ (- 1) \ mathbf (A)) ^ (n) = 0 \ mathrm (~ ~ or ~ ~) \ lim _ (n \ to \ infty) (\ mathbf (I) - \ mathbf (A) \ mathbf (X) ^ (- 1)) ^ (n) = 0) </Dd> </Dl> <Dd> lim n → ∞ (I − X − 1 A) n = 0 o r lim n → ∞ (I − A X − 1) n = 0 (\ displaystyle \ lim _ (n \ to \ infty) (\ mathbf (I) - \ mathbf (X) ^ (- 1) \ mathbf (A)) ^ (n) = 0 \ mathrm (~ ~ or ~ ~) \ lim _ (n \ to \ infty) (\ mathbf (I) - \ mathbf (A) \ mathbf (X) ^ (- 1)) ^ (n) = 0) </Dd> <P> then A is nonsingular and its inverse is </P>

Can you take determinant of non square matrix