4. 5. Let A be the following matrix: A = 5 -1 3 0 0 -5 -5 O 0 4 2 0 0 -5 -3 0 0 1 -1 1 1 0 0 0 0 Find all eigenvalues of A. For each eigenvalue, find a basis of its eigenspace. Show your work in details. Let A be the matrix in Problem 4. Is A invertible? If A is invertible, compute A-1. Show your work in details. Detailed justifications are required!

Sometimes symbols and abbreviations are used in place of a word or phrase. What word or phrase is represented or abbreviated by each of the following? a)post.b)c)d)e)pt.

Although Pythagoras preceded William Shakespeare by 2,000 years, the philosophy of the Pythagoreans is mentioned in Shakespears The Merchant of Venice. Here is a quote from that play: Thou almost makst me waver in my faith, To hold opinion will Pythagoras, That souls of animals infuse themselves Into the trunks of men. Research the Pythagoreans. What were the main beliefs held by their society? What part of the philosophy of the Pythagoreans was Shakespeare referring to specifically with this quote? What present-day religions share a similar belief? Write a paragraph or two about your findings.

In how many ways can 5 people stand in a line if one person demands to be first?

When can a theorem be cited as a reason reason in a proof?

In how many different ways can two letters be chosen from the letters A, B, C, D, E, F, and G? (The order of the two letters is not important.)