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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] and let $A = (a_{ij})$ be an $n \\times n$ [[square matrix|0JXZKX526R]] over $R$.\nThe matrix $A$ is *upper triangular* when $a_{ij} = 0$ for all $i \u003e j$.\nIt is *lower triangular* when $a_{ij} = 0$ for all $i \u003c j$.\nIt is *triangular* when it is upper triangular or lower triangular.\n\n## Remarks\n\nExample: $\\begin{pmatrix} 1 \u0026 2 \u0026 3 \\\\ 0 \u0026 4 \u0026 5 \\\\ 0 \u0026 0 \u0026 6 \\end{pmatrix}$ is triangular, because its entries below the diagonal, in positions $(2,1)$, $(3,1)$, $(3,2)$, are zero.\nNon-example: $\\begin{pmatrix} 1 \u0026 2 \\\\ 3 \u0026 4 \\end{pmatrix}$ is not triangular, because $a_{21} = 3 \\ne 0$ and $a_{12} = 2 \\ne 0$.\nA matrix is both upper and lower triangular exactly when it is diagonal.\nA triangular matrix is *strictly* triangular when its diagonal entries are also zero, and *unit* triangular when they are all $1$.\nSee also [[upper triangular matrix|0N4XY7Z3NN]] and [[lower triangular matrix|00J3KZP8Y0]].\n",
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