If the matrix \( \begin{bmatrix}z&x-y+2&x+4y-3\\x+y&0&a-3\\2x-y+3&a-1&0\end{bmatrix} \) is skew-symmetric, then the value of \( (a+x+y+z) \) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2
In a skew-symmetric matrix \( a_{ij}=-a_{ji} \), so every diagonal entry is 0. The diagonal gives
\( \displaystyle z=0 \)
From \( a_{12}=-a_{21} \):
\( \displaystyle x-y+2=-(x+y) \)
\( \displaystyle 2x+2=0 \)
\( \displaystyle x=-1 \)
From \( a_{13}=-a_{31} \):
\( \displaystyle x+4y-3=-(2x-y+3) \)
\( \displaystyle 3x+3y=0 \)
\( \displaystyle y=1 \)
From \( a_{23}=-a_{32} \):
\( \displaystyle a-3=-(a-1) \)
\( \displaystyle 2a=4 \)
\( \displaystyle a=2 \)
So
\( \displaystyle a+x+y+z=2-1+1+0 \)
\( \displaystyle =2 \)