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  "body": "\n## Summary\n\nThe generic matrix algebra is the algebra generated by finitely many $n \\times n$ matrices whose entries are independent commuting variables.\nIt is the \"most general\" algebra generated by $m$ matrices of size $n$.\nFor every commutative $k$-algebra $C$, the $k$-algebra generated by $m$ matrices in $M_n(C)$ is a quotient of it.\nThe quotient map substitutes elements of $C$ for the variables.\nOver an infinite field $k$, it is the relatively free algebra of the variety of algebras that satisfy all identities of $M_n(k)$.\nIt is a domain.\nIt is a basic object in the theory of [[polynomial identity rings|C0EDT81EGH]] and in the invariant theory of matrices.\n\n## Explanations\n\n### L1 human\n\nFill a $2 \\times 2$ matrix with four unknowns and a second one with four other unknowns.\nThe matrices you can build from these two by adding, multiplying and scaling form the generic matrix algebra.\nAny rule that holds in this algebra holds for all pairs of $2 \\times 2$ matrices, because you can substitute numbers for the unknowns.\nConversely, a rule that holds for all pairs of $2 \\times 2$ matrices over an infinite field holds here, so this algebra records exactly the identities of $2 \\times 2$ matrices.\n",
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