If matrix \(A = \begin{bmatrix}x & 2 & 3\\a & y & -5\\b & c & 0\end{bmatrix}\) is a skew-symmetric matrix, then
(A) \(x + y + c = 5\)
(B) \(c = 5\)
(C) \(a + b + c = 0\)
(D) \(a + b - c = 10\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (C) only
In a skew-symmetric matrix, \(a_{ij} = -a_{ji}\) for all i and j, so every diagonal entry is 0:
\(x = 0,\quad y = 0\)
Comparing each entry below the diagonal with the entry opposite it:
\(a = -2\)
\(b = -3\)
\(c = -(-5) = 5\)
So x = 0, y = 0, a = −2, b = −3 and c = 5.
Checking the statements
(A) \(x + y + c = 5\) — Correct. \(0 + 0 + 5 = 5\).
(B) \(c = 5\) — Correct. c is the negative of −5.
(C) \(a + b + c = 0\) — Correct. \(-2 - 3 + 5 = 0\).
(D) \(a + b - c = 10\) — Incorrect. \(-2 - 3 - 5 = -10\), not 10.