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
In a symmetric matrix, \(a_{ij} = a_{ji}\):
\(a_{23} = a_{32}\), so \(x = 3\)
In a skew-symmetric matrix, \(b_{ij} = -b_{ji}\), and every diagonal entry is 0:
\(b_{22} = 0\), so \(z = 0\)
\(b_{31} = -b_{13}\), so \(y = -(-10) = 10\)
Therefore
\(xy + yz + zx = 3 \times 10 + 10 \times 0 + 0 \times 3 = 30\)