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
Answer & explanation
Correct answer: option 2
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 $30$.