Question

show that every square matrix can be expressed as a sum of a symmetric and skew symmetric matrix. also show that if A and B are symmetric matrices of same order AB+BA is symeetric and AB-BA is a skew symmetric.

Answer 1

If a tilde over a matrix represents the transpose of that matrix

Any square matrix \(\textbf T\) can be written as
a sum of a Symmetric matrix \(\textbf S\) and a Skew-symmetric Matrix \(\textbf A\)
\[\textbf T= \textbf S+\textbf A\]

\[\textbf S=\frac{\textbf T+\widetilde{\textbf T} }{2}\qquad\qquad\textbf A=\frac{\textbf T-\widetilde{\textbf T} }{2}\]

Answer 2

i have never seen this kinda.
:(

Answer 3

what more we need to do here? @UnkleRhaukus

Answer 4

ive just done the first part

Answer 5

can you do all of it? @UnkleRhaukus

Answer 6

use the definition of \(\textbf S\)

Answer 7

what is that ?

Answer 8

did you read what i have wrote?