Express a matrix $A$ as the sum of a symmetric and a skew symmetric matrix

Question

I'm trying to express

$$A = \left[ \begin{array} { r r r } { 2 } & { - 2 } & { - 4 } \\ { - 1 } & { 3 } & { 4 } \\ { 1 } & { - 2 } & { - 3 } \end{array} \right]$$

as the sum of a symmetric and a skew symmetric matrix.

So far I have tried this:

converting into transpose

$A ^ { \prime } = \left[ \begin{array} { c c c } { 2 } & { - 1 } & { 1 } \\ { - 2 } & { 3 } & { 4 } \\ { - 4 } & { 4 } & { - 3 } \end{array} \right]$

well this dont seem to be symmteric matrix according to me. are there steps to be taken.

and for skew symmteric matrix i tried to do like this

transpose of equation $A ^ { \prime } = \left[ \begin{array} { c c c } { 2 } & { - 1 } & { 1 } \\ { - 2 } & { 3 } & { 4 } \\ { - 4 } & { 4 } & { - 3 } \end{array} \right]$

took minus as common $A ^ { \prime } = \left[ \begin{array} { c c c } { - 2 } & { 1 } & { - 1 } \\ { 2 } & { - 3 } & { 2 } \\ { 4 } & { - 4 } & { 3 } \end{array} \right]$

can you guide me how to proceed with them

Answer 1

HINT

Recall that for any square matrix $A$

$$A=\frac{A+A^T}{2}+\frac{A-A^T}{2}$$

and

$$\left(\frac{A+A^T}{2}\right)^T=\frac{A^T+A}{2}=\frac{A+A^T}{2}$$

$$\left(\frac{A-A^T}{2}\right)^T=\frac{A^T-A}{2}=-\frac{A-A^T}{2}$$