Question

Show that if A is an n × n matrix, then AA^T and A + A^T are symmetric

Answer 1

$Solution: \\ A ~is ~an~ n × n~ matrix~i.e. ~square~ matrix. \\If~ A^T=A ~then ~matrix ~ A~ is ~ symmetric. \\Let~K=AA^T \\\therefore K^T=(AA^T)^T \\~~~~~~~~~~~=(A^T)^TA^T~~~~~[Since~(AB)^T=B^TA^T] \\~~~~~~~~~~~=AA^T~~~~[Since ~(A^T)^T=A] \\~~~~K^T=K \\Hence ~ AA^T~is ~symmetric. \\Now ~let~us ~ consider ~C=A+A^T \\\therefore C^T=(A+A^T)^T \\~~~~~~~~~~~=A^T+(A^T)^T \\~~~~~~~~~~~=A^T+A~~~~~~~[Since~(A^T)^T=A ] \\~~~~~~~~~~~=A+A^T ~~~~~~~[A+A^T=A^T+A~~~Commutative Property] \\~~~~C^T=C \\Hence~ A+A^T~ is~ symmetric. \\Hence~if ~A~ is ~an~ n × n ~matrix,\\ then ~AA^T ~and~ A + A^T ~are~ symmetric.$