Question

A square matrix is called skew-symmetric if AT = -A. (a) (4 points) Explain why the main diagonal of a skew-symmetric matrix consists entirely of zeros. (b) (2 points) Provide examples of a 2 x 2 skew-symmetric matrix and a 3 x 3 skew-symmetric matrix. (6 points) Prove that if A and B are both n x n skew-symmetric matrices and c is a nonzero scalar, then A + B and cA are both skew-symmetric as well. (4 points) Find a basis for and the dimension of the subspace of all 2 x 2 skew-symmetric matrices.

Answer 1

Page 1 A square matrix is called skew-symmetric if AT = -A. (a) Explain why the main diagonal of a skew-symmetric matrix Consists entirely of zorous, let any A = (aij) nxn A is skew Symmetric & AT = -A (Qi;)-- (Qij) nxn for * (aji) nxn = -(ij) nxn a aji=-ai; Isijen diagonal entries we have izj aii =-all Isish 2010 20 Isish Jair zo isish Consists Hence diagonal of a skew - symmetric matrix entirely of zeros. Provide examples of a 2x2 Skew symmetric matrix Skew symmetric matrix. I (6) andazzz fo-2 but A= 2o AT = ( 2 ) = -A Hence A isskew symmetric matrix. 2019/10/19 01:59

Page 2 Be To I 27 let (-2 -4 oJ 3x3 87 -1 -27 7 BT =-B Hence B is Škew-symmetric matrix. (0) Prove that it A and B both are skew-symmetric matrix and c is a non-zero scalar then A+B and c A is skew Symmetric matrix. since A and B are skew symmetric matrix. a AT =- A and BT=-B. Consider (A+B)T= AT+BT = - A-B - (A+B) HAEL (ATB)T = -(ATB) Hence A+B is a skew-symmetric matrix. Consider (CAT = CAT = - CA (CAT= - CA Hence ca is a skew-symmetric matrix 2019/10/19 01:58

(d) find Page 3 Find the basis for and the dimension of all 2x2 skew -symmetric matrix. the basis of the subspace let A= (aisaxo such that A is skew-symmetric so A= (au aiz lazı azal By Part (as we know au 2922=0 so A Como Since A is skew-symmetric matrix a AT= -A lano Lazio azi=-012 ** *)-{ C lan is owned Az son ?.33. Simca :) an.com Hamas so the set & C I is the basts (2018/10/19 01:58 sin is non-zero Hence set Independent matrix and Dimension a l