Answer to Question #200935 in Linear Algebra for andisiwe

Question #200935

prove that A is a square matrix, then AAT and A + AT are symmetric


1
Expert's answer
2021-05-31T18:29:11-0400

A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, B is symmetric if BT=B,BB^T=B,B is a square matrix.


Transpose of a product (CD)T=DTCT(CD)^T=D^TC^T

The operation of taking the transpose is an involution (self-inverse): (BT)T=B(B^T)^T=B


The transpose respects addition: (C+D)T=CT+DT(C+D)^T=C^T+D^T


Then


(AAT)T=(AT)TAT=AAT(AA^T)^T=(A^T)^TA^T=AA^T

AATAA^T is symmetric.




(A+AT)T=AT+(AT)T=AT+A=A+AT(A+A^T)^T=A^T+(A^T)^T=A^T+A=A+A^T

A+ATA+A^T is symmetric.




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