# Homework Help: Linear Algebra - Matrix Symitry

1. Jul 22, 2012

### JeeebeZ

1. The problem statement, all variables and given/known data

Code (Text):
A matrix A is skew-symmetric if A[SUP]T[/SUP] = -A. Write the matrix
B below as the sum of a symmetric matrix and a skew-symmetric matrix.
B =  a b c
d e f
g h i
3. The attempt at a solution

So I'm Pretty sure that the
Symetric Matrix = B + BT
Skew Symetric Martix = B - BT

So B+BT+ B - BT should equal B but I get 2 B.

So B = 1/2((B+BT) + (B - BT))

Code (Text):

B[SUP]T[/SUP] = a d g
b e h
c f i

Sym = B + B[SUP]T[/SUP]
=  2a b+d c+g
d+b  2e f+h
g+c h+f  2i

Skew Sym = B - B[SUP]T[/SUP]
=  0  b-d c-g
d-b  0  f-h
g-c h-f  0

So If you add Those... You get

2a 2b 2c = 2B
2d 2e 2f
2g 2h 2i
Then divide by 2 and I'm done right?

2. Jul 22, 2012

### SammyS

Staff Emeritus
Actually, you're pretty close to getting a correct result.

The symmetric matrix you need is (1/2)(B + BT), the skew-symmetric matrix is (1/2)(B - BT) .

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