# Manuals/calci/PASCAL

**MATRIX("PASCAL",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

- The amazing relationship of these matrices are: .
- And its determinants also 1.i.e.,
- 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.

, where n=i+j, r=i.

- In other words,

.

- 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 |