Three types of matrices can be 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)
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(2)
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(3)
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The Pascal -matrix or order is implemented in the Wolfram Language as LinearAlgebra`PascalMatrix[n].
These matrices have some amazing properties. In particular, their determinants are all equal to 1
(4)
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and
(5)
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(Edelman and Strang).
Edelman and Strang give four proofs of the identity (5), the most straightforward of which is
(6)
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(7)
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(8)
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(9)
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where Einstein summation has been used.