Manuals/calci/PASCAL
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle order} 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
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_4 =\begin{pmatrix} 54 & 0 & 0 & 0 \\ 20 & 34 & 0 & 0 \\ 57 & 89 & -70 & 0 \\ 71 & -4 & -52 & 72 \\ \end{pmatrix}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_4 =\begin{pmatrix} 64 & 22 & -91 & -86 \\ 0 & 61 & 62 & -62 \\ 0 & 0 & 30 & -81 \\ 0 & 0 & 0 & -61 \\ \end{pmatrix}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_4 =\begin{pmatrix} 41 & 74 & 15 & -47 \\ 74 & -16 & 37 & 97 \\ 15 & 37 & 24 & -88 \\ -47 & 97 & -88 & -69 \\ \end{pmatrix}}
- The amazing relationship of these matrices are:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n = L_nU_n} .
- And its determinants also 1.i.e.,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |S_n|=|L_n|=|U_n|=1 }
- 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.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{ij} = {n \choose r} = \frac{n!}{r!(n-r)!},} , where n=i+j, r=i.
- In other words,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{ij} = _{i+j}{C}_{i} = \frac{(i+j)!}{(i)!(j)!}} .
- Here MATRIX("pascal") is showing the pascal matrix of order 3.
- So users can change the order of the matrix also.
Examples
- 1.MATRIX("pascal") =1
- 2.MATRIX("pascal",3)
| 1 | 1 | 1 |
| 1 | 2 | 3 |
| 1 | 3 | 6 |
- 3.MATRIX("pascal",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 |