If \( P=\begin{bmatrix}5+x & 3 & a\\b & x+y & c\\d & e & y+z\end{bmatrix} \) is a skew symmetric matrix, then which of the following are TRUE ?
(A) \( x+y+z=5 \)
(B) \( x+y+z=-5 \)
(C) \( a+d+e+c=0 \)
(D) \( z=3 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (C) only
In a skew-symmetric matrix every diagonal entry is 0 and \( p_{ji}=-p_{ij} \).
Diagonal entries:
\( \displaystyle 5+x=0\ \Rightarrow\ x=-5 \)
\( \displaystyle x+y=0\ \Rightarrow\ y=5 \)
\( \displaystyle y+z=0\ \Rightarrow\ z=-5 \)
Off-diagonal entries:
\( \displaystyle b=-3 \)
\( \displaystyle d=-a \)
\( \displaystyle e=-c \)
So x = −5, y = 5, z = −5, a + d = 0 and c + e = 0.
Checking the statements
(A) \( x+y+z=5 \) — Incorrect. x + y + z = −5 + 5 − 5 = −5.
(B) \( x+y+z=-5 \) — Correct. −5 + 5 − 5 = −5.
(C) \( a+d+e+c=0 \) — Correct. a + d = 0 and e + c = 0.
(D) \( z=3 \) — Incorrect. z = −5.