Here is one solution

Correct version:
symmetric true
for
row 1 to n do
for
column 1 to n do
if
a[row,column] != a[column,row] then
symmetric false
endif
endfor
endfor

because once the array is not symmetric ie symmetric= false it
cannot ever be symmetric even though subsequent comparisons may
indicate that it is!!!!!!
Another correct version is:
symmetric true
for
row 1 to n do
for
column 1 to n do
symmetric (symmetric) and
( a[row,column]=a[column,row])
endfor
endfor

It is not necessarily the only solution.