What is going on here is that MATLAB represents double precision numbers in IEEE 754 Double Precision format. That format involves a sign bit, several bits of base 2 exponent, and a number of bits of base 2 mantissa. Numbers are represented as
sign * 2^(exponent) * mantissa
where mantissa is an integer in the range 2^52 to 2^53-1
Notice that this is all binary. Multiplying by 10^20 involves a change of exponent and change of mantissa, not just a change of internal exponent.
When you multiply by 10^20, round-off occurs. When you divide again by 10^20, round-off occurs. The two round-offs do not exactly cancel.
The default display of numbers with "format long" is one digit short of displaying all of the digits needed to recreate a number exactly. You need to display the numbers using something like
fprintf('%.999g\n', A)
to see all of the digits.
