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
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 ❌