Question

Number of skew-symmetric matrices of order 3 whose elements are \( 0,0,0,1,-1,2,-2,3,-3 \) is (a) 8 (b) 12 (c) 12 (d) 48

Answer 1

\(A = \begin{pmatrix} 0 &x&x\\x&0&x\\x&x&0\end{pmatrix}\)

Number of skew symmetric matrices = 3! × 8 = 48.

[As, diagonal element must be 0and conjugate pair elements are additive inverse of each other in skew-symmetric matrix.

Aliter: 1 can be put by 6 ways

–1 can be put by 1 way

2 can be put by 4 ways

–2 can be put by 1 way

3 can be put by 2 ways

–3 can be put by 1 way

∴ Number of skew symmetric matrices = 6 × 1 × 4 × 1 × 2 × 1 = 48.