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

See Also

References