Let A be a nxxn matrix and let B=A+ATandC=A−AT (a) Show that B is symmetric...

Josalynn

Josalynn

Answered

2021-02-08

Let A be a nxxn matrix and let
B=A+ATandC=AAT
(a) Show that B is symmetric and C is skew symmetric.
(b) Show that every n × n matrix can be represented as a sum of a symmetric matrix and a skew-symmetric matrix.

Answer & Explanation

Nicole Conner

Nicole Conner

Expert

2021-02-09Added 97 answers

as we know that a matrix E is said to be symmetric if ET=E and skew symmetric if ET=E. 
as B=A+AT therefore, 
=BT=(A+AT)T 
=AT+(AT)T 
=AT+A(As(AT)T=A) 
=B 
as BT=B therefore, B is symmetric matrix. 
hence proved. 
As C=A−AT 
therefore, CT=(AAT)T 
=AT(AT)T 
=ATA 
=(AAT) 
=−C 
as CT=C therefore, C is skew symmetric matrix. Hence proved. 
Now we have to show that every nxxn matrix can be expressed as the sum of the symmetric and skew symmetric matrix. Let A be nxxn matrix. therefore, 
A=1/2(2A) 
=1/2(A+A) 
=1/2(A+AT+AAT) 
=1/2(B+C) 
where B=A+AT and B is a symmetric matrix and C=AAT and C is a skew symmetric matrix. 
herefore, 
A=1/2(B+C) 
=1/2(symmetric matrix +skew symmetric matrix) 
therefore, it has been showed that any matrix A of order nxxn can be expressed as the sum of symmetric and skew symmetric matrix.

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