To determine the average memory access time (AMAT) for the given computing system, we need to consider the hierarchical structure of the caches and their respective hit/miss rates and penalties. The formula for AMAT when considering multiple levels of cache is:

AMAT = L1 hit time + (L1 miss rate × (L2 hit time + L2 miss rate × memory access time))

Let's break this down into steps:

Step 1: Define the variables
L1 hit time = 1 ns (accessing data from L1 cache)
L1 hit rate = 90% = 0.9
L1 miss rate = 1 - 0.9 = 0.1
L2 hit time = 10 ns (miss penalty for L2 cache)
L2 hit rate = 80% = 0.8
L2 miss rate = 1 - 0.8 = 0.2
Main memory access time = 100 ns (miss penalty for main memory)

Step 2: Compute the effective L2 access time
For L2: Hit occurs 80% of the time, accounting for 10 ns. Miss incurs an additional 100 ns penalty:
Effective L2 access time = L2 hit time + L2 miss rate × memory access time
= 10 ns + 0.2 × 100 ns = 10 ns + 20 ns = 30 ns

Step 3: Compute the AMAT
Substitute the effective L2 access time into the AMAT formula:
AMAT = 1 ns + 0.1 × 30 ns
= 1 ns + 3 ns = 4 ns

Step 4: Verify the result
The computed AMAT is 4 ns, which falls within the expected range of 4 to 4 as specified. Thus, the solution is validated.