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  "body": "\n## Definition\n\nLet $R$ be a ring and let $n \\ge 0$ be an integer.\nA [[square matrix|0JXZKX526R]] $U = (u_{ij})$ of order $n$ over $R$ is *upper triangular* when $u_{ij} = 0$ for all $i \u003e j$.\nIt is *strictly upper triangular* when $u_{ij} = 0$ for all $i \\ge j$.\n\n## Remarks\n\nSums and products of upper triangular matrices of the same order are upper triangular.\nThe [[determinant of a triangular matrix|0K1KKMFHYH]] is the product of its diagonal entries.\nThe diagonal entries of an upper triangular matrix over a field are its [[eigenvalues|0BQAA2BK3Q]].\nA system $Ux = b$ with $U$ upper triangular and invertible is solved by [[back substitution|0ATFVE08MW]].\n\n## Explanations\n\n### L1 human\n\nAn upper triangular matrix has zeros everywhere below its diagonal.\nIts nonzero entries fill a triangle at the top right.\nFor example, $\\begin{pmatrix} 1 \u0026 2 \\\\ 0 \u0026 3 \\end{pmatrix}$ is upper triangular.\n",
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