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:

(A), (B) and (C) only

The correct answer is Option (2) → (A), (B) and (C) only

Given matrix: $A = \begin{bmatrix} x & 2 & 3 \\ a & y & -5 \\ b & c & 0 \end{bmatrix}$

Property of skew-symmetric matrix: $A^T = -A$, so diagonal elements = 0

Therefore, $x = 0$, $y = 0$, and $0$ (already) for third diagonal

Off-diagonal elements satisfy: $a_{ij} = -a_{ji}$

Check (2,1) element: $a = -2 \Rightarrow a = -2$

Check (3,1) element: $b = -3 \Rightarrow b = -3$

Check (3,2) element: $c = 5 \Rightarrow c = 5$

Now evaluate options:

(A) x+y+c = 0+0+5 = 5 ✅

(B) c = 5 ✅

(C) a+b+c = -2 + (-3) + 5 = 0 ✅

(D) a+b−c = -2 + (-3) − 5 = -10 ❌