Finding and creating Symmetric Matrix

Question

Given the matrix, find symmetric closure of it. I am having a hard time understanding how to solve this. \begin{bmatrix}1&0&1&0&0\\1&1&0&1&1\\0&0&0&0&0\\0&1&1&0&0\\1&0&0&1&1\end{bmatrix}

my try

[ 0 1 1 1 1

  1 0 1 0 1

  1 1 1 1 1

  1 0 1 1 1

   1 1 1 1 0]

Answer 1

Taking the symmetric closure of a relation just means adding the smallest set of ordered pairs to the relation so that the result is symmetric. So for the matrix representing the relation, this is really equivalent to adding the fewest number of $1$'s so that the resulting matrix is symmetric, i.e. setting $A_{i,j} = 1$ wherever $A_{j, i} = 1$. This yields:

\begin{bmatrix} 1&1&1&0&1\\ 1&1&0&1&1\\ 1&0&0&1&0\\ 0&1&1&0&1\\ 1&1&0&1&1 \end{bmatrix}