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  "body": "\n## Definition\n\nLet $R$ be a ring and let $A = (a_{ij})$ and $B = (b_{ij})$ be $m \\times n$ [[matrices|0VD9ZSWWJG]] over $R$.\nThe *sum* $A + B$ is the $m \\times n$ matrix with entries\n$$(A + B)_{ij} = a_{ij} + b_{ij}$$\nfor $1 \\le i \\le m$ and $1 \\le j \\le n$.\n\n## Remarks\n\nThe sum is defined only for two matrices of the same size.\nMatrix addition is associative and commutative, the [[zero matrix|0ZTKJJDGR7]] is its identity element, and $-A = (-a_{ij})$ is the additive inverse of $A$.\nThus $M_{m \\times n}(R)$ is an abelian group under addition.\nFor example, $\\begin{pmatrix} 1 \u0026 2 \\\\ 3 \u0026 4 \\end{pmatrix} + \\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix} = \\begin{pmatrix} 1 \u0026 3 \\\\ 4 \u0026 4 \\end{pmatrix}$.\n\n## Explanations\n\n### L1 human\n\nTo add two matrices of the same size, add the numbers in the same position.\nThe result has the same size as the two matrices.\nMatrix addition has the same rules as the addition of numbers.\n",
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