A invertible⇔ det(A)≠0⇔ rank(A)=n⇔ A has n pivot positions⇔ Ax=0 has only x=0⇔ A is row equivalent to In⇔ columns of A are linearly independent⇔ A is bijectiveA\ \text{invertible} \\ \Leftrightarrow\; \det(A) \neq 0 \\ \Leftrightarrow\; \text{rank}(A) = n \\ \Leftrightarrow\; A\ \text{has}\ n\ \text{pivot positions} \\ \Leftrightarrow\; A x = 0\ \text{has only}\ x = 0 \\ \Leftrightarrow\; A\ \text{is row equivalent to}\ I_n \\ \Leftrightarrow\; \text{columns of}\ A\ \text{are linearly independent} \\ \Leftrightarrow\; A\ \text{is bijective}A invertible⇔det(A)=0⇔rank(A)=n⇔A has n pivot positions⇔Ax=0 has only x=0⇔A is row equivalent to In⇔columns of A are linearly independent⇔A is bijectiveGaussian Elimination - O(n3)O(n^3)O(n3) human handMatrix Decomposition - O(n3)O(n^3)O(n3) code implementation usually with LU decomposition Adjugate matrix - O(n4)O(n^4)O(n4)Inverse Matrix NotionInvertible Matrix TheoremAdjugate matrix
