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<div style="font-size:30px">'''PASCAL'''</div><br/>
<div style="font-size:30px">'''MATRIX("PASCAL",order)'''</div><br/>
*<math>order</math> is the size of the Pascal matrix.
==Description==
*This function returns the matrix of any order with the property of Pascal.
*The Pascal matrix is an infinite matrix containing the binomial coefficients as its elements.
*To obtain a pascal matrix there are three ways: as either an upper-triangular matrix(U), a lower-triangular matrix(L), or a symmetric matrix(S).
*Example for these matrices are:
<math>L_4 =\begin{pmatrix}
54 & 0 & 0 & 0 \\
20 & 34 & 0 & 0 \\
57 & 89 & -70 & 0 \\
71 & -4 & -52 & 72 \\
\end{pmatrix}</math>
<math>U_4 =\begin{pmatrix}
64 & 22 & -91 & -86 \\
0 & 61 & 62 & -62 \\
0 & 0 & 30 & -81 \\
0 & 0 & 0 & -61 \\
\end{pmatrix}</math>
<math>S_4 =\begin{pmatrix}
41 & 74 & 15 & -47 \\
74 & -16 & 37 & 97 \\
15 & 37 & 24 & -88 \\
-47 & 97 & -88 & -69 \\
\end{pmatrix}</math>
*The amazing relationship of these matrices are:<math>S_n = L_nU_n</math>.
*And its determinants also 1.i.e.,<math> |S_n|=|L_n|=|U_n|=1 </math>
*The Pascal matrix can actually be constructed by taking the matrix exponential of a special subdiagonal or superdiagonal matrix.
*The elements of the symmetric Pascal matrix are the binomial coefficients, i.e.
<math>S_{ij} = {n \choose r} = \frac{n!}{r!(n-r)!},</math>, where n=i+j, r=i.
*In other words,
<math>S_{ij} = _{i+j}{C}_{i} = \frac{(i+j)!}{(i)!(j)!}</math>.
*Here MATRIX("pascal") is showing the pascal matrix of order 3.
*So users can change the order of the matrix also.