If \(\begin{bmatrix}x-1 & y-2 & z-3\\3 & u-4 & v-5\\8 & -4 & w-6\end{bmatrix}\) is a skew-symmetric matrix, then xu + yv + zw is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → −35
In a skew-symmetric matrix, \(a_{ji} = -a_{ij}\), so every diagonal entry is 0:
\( x - 1 = 0,\quad u - 4 = 0,\quad w - 6 = 0 \)
\( x = 1,\quad u = 4,\quad w = 6 \)
The entries above the diagonal are the negatives of those below it:
\( y - 2 = -3,\quad y = -1 \)
\( z - 3 = -8,\quad z = -5 \)
\( v - 5 = -(-4) = 4,\quad v = 9 \)
Therefore
\( xu + yv + zw = (1)(4) + (-1)(9) + (-5)(6) \)
\( = 4 - 9 - 30 = -35 \)