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Pascal matrix
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<p> In <a href="/facts/Matrix_theory/qa8FI4ko">matrix theory</a> and <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, a Pascal matrix is a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> (possibly <a href="/facts/Matrix_(mathematics)/qa8FI4ko">infinite</a>) containing the <a href="/facts/Binomial_coefficient/Tsx7eY5h">binomial coefficients</a> as its elements. It is thus an encoding of <a href="/facts/Pascal%27s_triangle/tb6BKXbc">Pascal's triangle</a> in matrix form. There are three natural ways to achieve this: as a <a href="/facts/Lower-triangular_matrix/qjXSFMM3">lower-triangular matrix</a>, an <a href="/facts/Upper-triangular_matrix/qjXSFMM3">upper-triangular matrix</a>, or a <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrix</a>. For example, the 5 × 5 matrices are: </p><p> L 5 = ( 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 ) {\displaystyle L_{5}={\begin{pmatrix}1&0&0&0&0\\1&1&0&0&0\\1&2&1&0&0\\1&3&3&1&0\\1&4&6&4&1\end{pmatrix}}\,\,\,} U 5 = ( 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 ) {\displaystyle U_{5}={\begin{pmatrix}1&1&1&1&1\\0&1&2&3&4\\0&0&1&3&6\\0&0&0&1&4\\0&0&0&0&1\end{pmatrix}}\,\,\,} S 5 = ( 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70 ) = L 5 × U 5 {\displaystyle S_{5}={\begin{pmatrix}1&1&1&1&1\\1&2&3&4&5\\1&3&6&10&15\\1&4&10&20&35\\1&5&15&35&70\end{pmatrix}}=L_{5}\times U_{5}} There are other ways in which Pascal's triangle can be put into matrix form, but these are not easily extended to infinity. </p>