Math 110 Linear Algebra Fall 2009 Haiman Problem Set 9 Due 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 1 AQ det A Deduce that if V is a finite dimensional vector space and T V V is a linear transformation then det T does not depend on the choice of the ordered basis of V 2 A matrix of the form 1 x1 x21 xn 1 1 1 x2 x2 xn 1 2 2 A n 1 2 1 xn xn xn is called a Vandermonde matrix a Show that the determinant det A is a polynomial in the variables x1 x2 xn in which every term has degree n n 1 2 The degree of a monomial xa11 xa22 xann is defined to be a1 an b Show that det A becomes zero if xi xj for any i and j This implies that det A is divisible as a polynomial in the xi s by the product Y xj xi 1 i j n in det A is equal to 1 c Show that the coefficient of the monomial x01 x12 xn 1 n 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 form A B M 0 C where A and C are square Express det M in terms of det A and det C Give reasoning to justify your answer 4 Prove that an upper triangular matrix that is a square matrix A such that aij 0 for 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 rows are held constant Suppose f also has the property that f A 0 whenever A has two equal rows Prove that f B f A whenever B is obtained from A by switching two rows 6 A permutation of order n is a bijective function 1 n 1 n If is a permutation of order n we define the permutation matrix P to be the n n matrix with 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 sends ej to e j b Show that LP sends x 1 x n T to x1 xn T c The inversion number i is defined to be the number of pairs of integers 1 j k n such that j k Prove that det P 1 i
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