Math 110—Linear AlgebraFall 2009, HaimanProblem Set 9Due Monday, Nov. 2 at the beginning of lecture.1. Prove that if A and Q are n×n matrices over F, with Q invertible, then det(Q−1AQ) =det(A). Deduce that if V is a finite-dimensional vector space and T : V → V is a lineartransformation, then det([T ]β) does not depend on the choice of the ordered basis β of V .2. A matrix of the formA =1 x1x21. . . xn−111 x2x22. . . xn−12............1 xnx2n. . . xn−1nis called a Vandermonde matrix.(a) Show that the determinant det(A) is a polynomial in the variables x1, x2, . . . , xninwhich every term has degree n(n − 1)/2. (The degree of a monomial xa11xa22· · · xannis definedto be a1+ · · · + an.)(b) Show that det(A) becomes zero if xi= xjfor any i and j. This implies that det(A)is divisible as a polynomial in the xi’s by the productY1≤i<j≤n(xj− xi).(c) Show that the coefficient of the monomial x01x12· · · xn−1nin det(A) is equal to 1.(d) Deduce from the above that det(A) is equal to the product in part (b).3. Suppose M is an n × n matrix of the formM =A B0 Cwhere A and C are square. Express det(M) in terms of det(A) and det(C). Give reasoningto justify your answer.4. Prove that an upper triangular matrix (that is, a square matrix A such that aij= 0for j < i) is invertible if and only if all its diagonal entries are non-zero.5. Suppose f : Mm×n(F) → F is an m-multilinear function of the rows of A ∈ Mm×n(recall that this means f is linear as a function of each row separately when the other rowsare held constant). Suppose f also has the property that f(A) = 0 whenever A has twoequal rows. Prove that f(B) = −f(A) whenever B is obtained from A by switching tworows.6. A permutation of order n is a bijective function π : {1, . . . , n} → {1, . . . , n}. If π is apermutation of order n, we define the permutation matrix P (π) to be the n × n matrix with(π(j), j)-th entry equal to 1 for all j = 1, . . . , n, and all other entries equal to zero.(a) Show that the linear transformation LP (π)sends ejto eπ(j).(b) Show that LP (π)sends (xπ(1), . . . , xπ(n))Tto (x1, . . . , xn)T.(c) The inversion number i(π) is defined to be the number of pairs of integers 1 ≤ j <k ≤ n such that π(j) > π(k). Prove that det(P (π)) =
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