For any positive integer n, the MATLAB command P = pascal(n) will generate an n n matrix

Chapter 6, Problem 25

(choose chapter or problem)

For any positive integer n, the MATLAB command P = pascal(n) will generate an n n matrix P whose entries are given by pi j = _ 1 if i = 1 or j = 1 pi1, j + pi, j1 if i > 1 and j > 1 The name pascal refers to Pascals triangle, a triangular array of numbers that is used to generate binomial coefficients. The entries of the matrix P form a section of Pascals 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) In part (a) we saw that the determinant of the 6 6 Pascal matrix is 1, but if we subtract 1 from the (6, 6) entry, the matrix becomes singular. Will this happen in general for n n Pascal matrices? To answer this question, consider the cases n = 4, 8, 12. In each case, set P = pascal(n) and compute its determinant. Next, subtract 1 from the (n, n) entry and compute the determinant of the resulting matrix. Does the property that we discovered in part (a) appear to hold for Pascal matrices in general? (c) 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? (d) Set R(8, 8) = 0 and Q = R_ 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.