Question

[tex] Obtain X and Y if X+Y=A\left[\begin{array}{ccc}1&6&7\\2&5&9\\3&4&8\end{array}\right] ,Where X is a symmetric and Y is a skew-symmetric matrix. [\tex]

Answer 1

Answer:

$X=\frac{1}{2}\left[\begin{array}{ccc}2&8&10\\8&10&13\\10&13&16\end{array}\right]$

$Y=\frac{1}{2}\left[\begin{array}{ccc}0&4&4\\-4&0&5\\-4&-5&0\end{array}\right]$

Step-by-step explanation:

$X+Y=A$

$A=\left[\begin{array}{ccc}1&6&7\\2&5&9\\3&4&8\end{array}\right]$

$A^T=\left[\begin{array}{ccc}1&2&3\\6&5&4\\7&9&8\end{array}\right]$

Then

$X=\frac{1}{2}(A+A^T)$

$X=\frac{1}{2}(\left[\begin{array}{ccc}1&6&7\\2&5&9\\3&4&8\end{array}\right]+\left[\begin{array}{ccc}1&2&3\\6&5&4\\7&9&8\end{array}\right])$

$X=\frac{1}{2}\left[\begin{array}{ccc}2&8&10\\8&10&13\\10&13&16\end{array}\right]$

$\text{since }X=X^T,\text{ X is symmetric matrix}$

$Y=\frac{1}{2}(A-A^T)$

$Y=\frac{1}{2}(\left[\begin{array}{ccc}1&6&7\\2&5&9\\3&4&8\end{array}\right]-\left[\begin{array}{ccc}1&2&3\\6&5&4\\7&9&8\end{array}\right])$

$Y=\frac{1}{2}\left[\begin{array}{ccc}0&4&4\\-4&0&5\\-4&-5&0\end{array}\right]$

$\text{since }Y=-Y^T,\text{ Y is skew symmetric matrix}$