Suppose that $A = B+C$ where, $B = B^T$ and $C = -C^T$. If $(CB)^T = D$ and $D = [ d_{ij} ]_{3*3}$, what is $d_{12}+d_{21}+d_{13}+d_{31}$?

Question

The question had already given the matrix A.

$A$ = $\begin{bmatrix} 2 & -2 & -4\\ -1 & 3 & 4\\ 1 & -2 & -3 \end{bmatrix}$

I can't figure out what are $B$ and $C$.

It's obvious that $D = (CB)^T = B^T$ $C^T = B (-C)$

But I don't know how to find $B$ and $C$ from $A = B+C$

I'm honestly lost this. Could somebody help?

Answer 1

Guide:

Use the decomposition

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

Let $B=\frac{A+A^T}{2}$ and $C=\frac{A-A^T}{2}$

Edit:

The formula is obtained from $$A=B+C$$ $$A^T=B^T+C^T$$

adding the two equations up, we have

$$B+B^T=2B=A+A^T$$ subtracting the two equations, you can solve for $C$.