For any positive integer n, the MATLAB command P = pascal(n) will generate an n × n matrix P whose entries are given by

The name pascal refers to Pascal's triangle, a triangular array of numbers that is used to generate binomial coefficients. The entries of the matrix P form a section of Pascal's triangle.
(a) Set
P = pascal(6)
and compute the value of its determinant. Now subtract 1 from the (6, 6) entry of P by setting
P(6, 6) = P(6, 6) - 1
and compute the determinant of the new matrix P. What is the overall effect of subtracting 1 from the (6, 6) entry of the 6 × 6 Pascal matrix?
(b) Set
P = pascal(8)
and examine its leading principal submatrices. Assuming that all Pascal matrices have determinants equal to 1, why must P be positive definite? Compute the upper triangular Cholesky factor R of P. How can the nonzero entries of R be generated as a Pascal triangle? In general, how is the determinant of a positive definite matrix related to the determinant of one of its Cholesky factors? Why must det(P) = 1?
(c) Set
R(8, 8) = 0 and Q = R€² * R
The matrix Q should be singular. Why? Explain. Why must the matrices P and Q be the same except for the (8, 8) entry? Why must q88 = p88 - 1? Explain. Verify the relation between P and Q by computing the difference P - Q.

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