The number of different n x n symmetric matrices with each element being either 0 or 1 is: (Note: power(2,n) is same as ...

Question

The number of different n x n symmetric matrices with each element being either 0 or 1 is: (Note: power(2,n) is same as 2^n)
A. power(2,n)
B. power(2,n^2)
C. power(2,(n^2+n)/2)
D. power(2,(n^2-n)/2)

Accepted Answer

power(2,(n^2+n)/2). Answer: C. A symmetric matrix satisfies A[i][j] = A[j][i]. So entries above the diagonal are mirror images of entries below. Only the diagonal and the upper triangle can be chosen freely. For an n x n matrix this gives n diagonal entries plus n*(n-1)/2 upper-triangle entries, totalling (n^2+n)/2 free entries, each taking value 0 or 1.

The number of different n x n symmetric matrices with each element being either 0 or 1 is: (Note: power(2,n) is same as 2^n) A. power(2,n) B. power(2,n^2) C. power(2,(n^2+n)/2) D. power(2,(n^2-n)/2)