Question

2. Let A be an n x n matrix with AT =-A (a) Prove that A has value 0. (b) Prove that A has determinant 0 if n is odd.

Answer 1

0 let Abe an aon matrin with AT =-A - A is skew symmetric ATS-A & A is sxew symmetric aija-aji & ith yow and jth column. (a) prove A has claim:- that value o. then proof: – HA is skew symmetric matoin air = - qji & values of i and j the diagonal elements of a skew symmetric matrix are equal to zero. the diagonal elements of a are characterized by ais where izj it i- j then aij = ari = 9;3 it is sken symmetrice then Qji - - 9jj qui a – aii daci = 0 air = 0

a matrix is the product of air the value of but air=0 therefore, value of A=0 (6) to 111=0 is odd. we need prove that it an suppose that is an odd integer and proof: - matrix . let symmetric A be an non skew by AT = -A definition of Thus , we have skew symmetric. det (A) = det (AT) { ifa is any nxn mateix then delat = detd {.

det(A) = det (AT) = det(A) (since 17 =-^) = (-1) "det (A) della) = det (a).(-1) -det(a) (sina, nisool) det(A) = - det (A) -) 2 det (A) = 0 or detA=0 det A =