If \(A =\begin{bmatrix}x+z&2&-3\\x&0&4\\3&x-y&0\end{bmatrix}\) is a skew-symmetric matrix, then which of the following are true?
(A) \(y \gt z \gt x\)
(B) \(x \gt y\)
(C) \(x + y + z \gt 0\)
(D) \(z \gt x\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (C) and (D) only
In a skew-symmetric matrix, \(a_{ji} = -a_{ij}\), so every diagonal entry is 0.
\(a_{21} = -a_{12}:\quad x = -2\)
\(a_{11} = 0:\quad x + z = 0,\quad z = 2\)
\(a_{32} = -a_{23}:\quad x - y = -4,\quad y = 2\)
(The pair \(a_{13} = -3,\ a_{31} = 3\) also agrees.)
So x = −2, y = 2 and z = 2.
Checking the statements
(A) \(y \gt z \gt x\) — Incorrect. y = z = 2, so y > z fails.
(B) \(x \gt y\) — Incorrect. −2 < 2.
(C) \(x + y + z \gt 0\) — Correct. \(-2 + 2 + 2 = 2 \gt 0\).
(D) \(z \gt x\) — Correct. 2 > −2.