Problem 1.  Given a matrix 

  where a is a real numnber. Find the rank of A1 and give the corresponding range of a.

(a) [3%] Assume the miss rate of an instruction cache is 2% and the miss rate of the data cache is 4%, If a processor has a CPI of 2 without any memory stalls and the miss penalty is 200 cycles for all misses. Assume the frequency of all loads and stores is 36%. Determine how much faster a processor would run with a perfect cache that never missed.

(b) [3%] If the processor has a base CPI of 1, and clock rate of 4 GHz. Assume the memory access time is 100 ns, including al the miss handling. Suppose the miss rate per instruction at the primary cache is 3%. Now we add a secondary data cache that has a 5 ns access time for either a hit or a miss, and is large enough to reduce the miss rate to main memory to 0.5%. What is the total CPI of this two-level cache?

(a) [3%] For a single-level page table, how much physical memory is needed for storing the page table?

(b) [3%] In a memory hierarchy system, we can integrate virtual memory, TLB and cache. A memory reference can encounter three different types of misses: a TLB miss, a page fault and a cache miss. Considering all the combinations of these three events with one or more occurring. Identify the number for the three events which is impossible.

(i)  Find the Eigenvalues and Eigenvectors for.

(ii) Show that there exists a matrix , which is similar to the matrix , in the Jordon form, i.e., 

(i) Find the range space (i.e., column space) of A, denoted as R(A), and the row space of A (i.e., R()) where denotes the transpose of A;

(ii) find the projection matrix such that , and;

(ii) find the eignevalues and eigenvectors of .

(i) Is Tr(AX)≥ 0 true?