If $A =\begin{bmatrix}2&-3&4\\-3&5&x\\4&3&0\end{bmatrix}$ is a symmetric matrix and $B =\begin{bmatrix}0&2&-10\\-2&z&6\\y&-6&0\end{bmatrix}$ is a skew-symmetric matrix, then the value of $(xy + yz + zx)$, is

30

The correct answer is Option (2) → 30

$A=\begin{pmatrix}2 & -3 & 4\\ -3 & 5 & x\\ 4 & 3 & 0\end{pmatrix}$ is symmetric.

Symmetric condition gives:

$a_{23}=a_{32}\Rightarrow x=3$

$B=\begin{pmatrix}0 & 2 & -10\\ -2 & z & 6\\ y & -6 & 0\end{pmatrix}$ is skew–symmetric.

Skew–symmetric condition gives:

$b_{12}=-b_{21}\Rightarrow 2=2$ (already fine)

$b_{13}=-b_{31}\Rightarrow -10=-y\Rightarrow y=10$

$b_{23}=-b_{32}\Rightarrow 6=6$ (already fine)

$b_{22}=0\Rightarrow z=0$

Now compute $xy+yz+zx$:

$x=3,\;y=10,\;z=0$

$xy+yz+zx=(3)(10)+(10)(0)+(0)(3)=30$

The value of $(xy+yz+zx)$ is $30$.