A pascal matrix is a two-dimensional square matrix holding numbers from [[Pascal's triangle]], also known as [[Evaluate binomial coefficients|binomial coefficients]] and which can be shown as <big><sup>n</sup>C<sub>r</sub>.</big>
Shown below are truncated 5-by-5 matrices M[i, j] for i,j in range 0..4. <br>
A Pascal upper-triangular matrix that is populated with <big><sup>j</sup>C<sub>i</sub>:</big>
<pre>
[[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]]
</pre>
A Pascal lower-triangular matrix that is populated with <big><sup>i</sup>C<sub>j</sub></big> (the transpose of the upper-triangular matrix):
<pre>
[[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]]
</pre>
A Pascal symmetric matrix that is populated with <big><sup>i+j</sup>C<sub>i</sub>:</big>
<pre>
[[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]]
</pre>
;Task:
Write functions capable of generating each of the three forms of n-by-n matrices.
Use those functions to display upper, lower, and symmetric Pascal 5-by-5 matrices on this page.
The output should distinguish between different matrices and the rows of each matrix (no showing a list of 25 numbers assuming the reader should split it into rows).
;Note:
The [[Cholesky decomposition]] of a Pascal symmetric matrix is the Pascal lower-triangle matrix of the same size.
