Question

Let A be the set of all $4\times 4$ skew symmetric matrices whose entries are $-1,0or1$. If there are exactly four 0s, six 1s and six -1s then the number of such matrices is

• A. $64$
• B. $32$
• C. $0$
• D. $16$

Answer 1

The correct option is A $64$

A is a skew-symmetric matrix of order 4X4.
For skew symmetric matrix $a_{ij}=-a_{ji}$
So,let $A=\left[\begin{array}{cccc}0& a& b& c\\ -a& 0& d& e\\ -b& -d& 0& f\\ -c& -e& -f& 0\end{array}\right]$
Each element above diagonal can be filled in 2 ways (1 or -1). And once an element above diagonal is filled, one element below will be negative of it.
$\therefore$ total number of ways$=2^6\times 1^6=64$ ways .
($\because$ there are $6$ elements above the diagonal and $6$ below.)