A Pascal matrix is any of three types of 
matrices obtained by writing
Pascal's triangle as a lower
triangular matrix and truncating appropriately: a symmetric
matrix 
with 
,
a lower triangular matrix 
with 
, and an upper
triangular matrix 
with 
,
where 
,
1, ..., 
.
For example, for 
,
these would be given by
 |  | ![[1 1 1 1; 1 2 3 4; 1 3 6 10; 1 4 10 20]](/images/equations/PascalMatrix/Inline13.svg) |
(1)
|
 |  | ![[1 ; 1 1 ; 1 2 1 ; 1 3 3 1]](/images/equations/PascalMatrix/Inline16.svg) |
(2)
|
 |  | ![[1 1 1 1; 1 2 3; 1 3; 1].](/images/equations/PascalMatrix/Inline19.svg) |
(3)
|
All three are integer matrices and hence rational matrices.
The Pascal 
-matrix
of order 
is implemented in the Wolfram Language
as LinearAlgebra`PascalMatrix[n].
Their determinants satisfy
 |
(4)
|
and
 |
(5)
|
(Edelman and Strang).
Edelman and Strang give four proofs of the identity (5), the most straightforward of which is
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
where Einstein summation has been used.
