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  "body": "\n## Summary\n\nA generalized inverse of a matrix $A$ is a matrix $G$ with $AGA = A$.\nEvery matrix over a field has one, even when it is not square or not invertible.\nGeneralized inverses give solutions of consistent linear systems $Ax = b$ in the form $x = Gb$.\nThe [[Moore–Penrose inverse|0YY8VX28JR]] is a generalized inverse with three more conditions that make it unique.\nThe [[Drazin inverse|9FB0AZSEGM]] is a related kind of inverse for square matrices, though it does not satisfy $AGA = A$ in general.\n\n## Explanations\n\n### L1 human\n\nOnly some square matrices can be inverted.\nA generalized inverse keeps one useful property of an inverse: applied in between, it does not change $A$.\nThink of $G$ as a map that sends each vector in the image of $A$ back to one of its preimages.\nFor example, for $A = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix}$, every matrix $G = \\begin{pmatrix} 1 \u0026 a \\\\ b \u0026 c \\end{pmatrix}$ satisfies $AGA = A$.\nSo a generalized inverse exists but is usually not unique.\n",
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