Show that A + A^T is symmetric?

Question

hey everyone, im having trouble proving this, my book had a short explanation but im still a bit lost.
this is matrix A = and i found that A^T is
(A)row1= [ 2 -3 -5] (A^T)row1=[2 -1 1]
(A)row2= [ -1 4 5] (A^T)row2=[-3 4 -3]
(A)row3 =[1 -3 -4] (A^T)row3=[-5 5 -4]
this is the best way i can present these. A is a3x3 matrix and so is its transposed verison.
the question was how do i show that A + A^T is symmetric. and the following question gives a definition: an nxn matrix A is a skew- symmetric if A^T=-A. using the orignial A matrix above how do i show that A-A^T is Skew Symmetric? please explain how do i find these answers so i can have an understanding of the question. i appreciate any help :)

Answer 1

a b c

d e f

g h i

+

a d g

b e h

c f i

=

a+a b+d c+g

d+g e+e f+h

g+c h+f i+i



Because addition is "abelian",

b+d = d+b



this means that the value of cell(1,2) = cell(2,1)

and so on for each position

Answer 2

Its been years since I reviewed linear algebra, but the best way to approach a problem like this is to construct a small 3X3 matrix marking off each entry as Aij in the original matrix and then writing out the transverse then adding.



Note, for symmetry you can ignore the diagonal elements.



Let the sum of the two matrices be called matrix C, with the Cij for each entry.



Pick any non-diagonal element, say Cij, What is it the sum of? It is Aij + Aji

What is the value of Cji, that is what is it the sum of ? It is Aji + Aij.



Since we still can apply the communative laws of addition each non-diagonal element is equal. Therefore the sum matrix C is symmetric

Answer 3

A matrix B is symmetric if B = B^T

Correct?



Does (A + A^T) = (A + A^T)^T ??



Hypothesis:

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



By rule, the Transpose of a sum is the sum of the transpose.

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



The transpose of a transpose of A is A

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



Matrix addition is associative.

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



Seems to be the case...



I hope this is sufficient proof.

Answer 4

Go back to the definition of symmetric and transpose, and write out the (i,j) entry of your matrix symbolically.