Question

The number of distinct $5\times 5$ symmetric matrices with each element being $0$ or $1$ is

• A. ${15}^2$
• B. ${25}^2$
• C. $2^{15}$
• D. $2^{25}$

Answer 1

The correct option is C $2^{15}$

For a symmetric matrix $A,$
$a_{ij}=a_{ji}\forall i,j$
which implies that we can change the elements upper to the diagonal and lower will be fixed or vice-versa.
So, total elements $=1+2+3+\cdots +n=\frac{n(n+1)}{2}=\frac{5\times 6}{2}=15$
Now, there are only $2$ possibilities, $0$ and $1$ for $a_{ij}.$
$\therefore$ Total number of symmetric matrix $=2^{15}$