Wikipedia states that: If matrix A is invertible, then it admits an LU (or LDU) factorization if and only if all its leading principal minors are nonzero.
However matrix \begin{bmatrix}0&1\\1&1\end{bmatrix} has leading principal minors: M1 = 0 and M2 = -1 and this matrix has LU factorization: \begin{equation} \begin{bmatrix}1&0\\0&1\end{bmatrix} \begin{bmatrix}1&1\\0&1\end{bmatrix} \end{equation} When I made a mistake using this logic?