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  "body": "\n## Definition\n\nLet $R$ be a ring.\nLet $m = m_1 + \\cdots + m_p$ and $n = n_1 + \\cdots + n_q$ with positive integers $m_i$ and $n_j$.\nFor each $i \\le p$ and $j \\le q$, let $A_{ij}$ be an $m_i \\times n_j$ [[matrix|0VD9ZSWWJG]] over $R$.\nThe *block matrix* with blocks $A_{ij}$ is the $m \\times n$ matrix\n$$A = \\begin{pmatrix} A_{11} \u0026 \\cdots \u0026 A_{1q} \\\\ \\vdots \u0026 \\ddots \u0026 \\vdots \\\\ A_{p1} \u0026 \\cdots \u0026 A_{pq} \\end{pmatrix},$$\nin which $A_{ij}$ fills rows $m_1 + \\cdots + m_{i-1} + 1$ to $m_1 + \\cdots + m_i$ and columns $n_1 + \\cdots + n_{j-1} + 1$ to $n_1 + \\cdots + n_j$.\n\n## Remarks\n\nIf the column partition of $A$ matches the row partition of $B$, then $AB$ is computed block by block: $(AB)_{ik} = \\sum_j A_{ij} B_{jk}$.\nIn Mathlib, `Matrix.fromBlocks` makes the $2 \\times 2$ block matrix $\\begin{pmatrix} A \u0026 B \\\\ C \u0026 D \\end{pmatrix}$.\nThe [[determinant of a block triangular matrix|0J9TVTNB4K]] is the product of the determinants of its diagonal blocks.\n\n## Explanations\n\n### L1 human\n\nA block matrix is a large matrix cut into smaller rectangles.\nWe can treat each rectangle like a single entry.\nThen we add and multiply block matrices with the usual rules, as long as the sizes fit.\n",
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