Let A ∈ Mat n×n(R) be a real square matrix.

(a) Suppose that A is symmetric, positive semi-definite, and orthogonal. Prove that A is the identity matrix.

(b) Suppose that A satisfies A = −A^T . Prove that if λ ∈ C is an eigenvalue of A, then λ¯ = −λ.

From now on, we assume that A is idempotent, i.e. A^2 = A.

(c) Prove that if λ is an eigenvalue of A, then λ is equal to 0 or 1.

(d) Set V1 = {v ∈ C n | Av = v} and V0 = {v ∈ C n | Av = 0}. Show that im A = V1 and ker A = V0.

(e) Prove that A is diagonalizable.

You're cleaning up your little nephew's toy room. There are T toys on the floor and n empty toy storage boxes. You randomly throw toys into boxes, and when you're done the box with the most toys contains N toys.

(a)What is the smallest that N could be when T=2n+1?

(b) What is the smallest that N could be when T=kn+1?

(c)Now suppose that the number of toys T satisfies

T<n(n−1)/2.

Prove that when you are done cleaning there will be (at least) one pair of boxes that contain the same number of toys.

Consider the weighted voting system

​[15​: 12​, 9​, 5​, 2​].

Let A = 2 1 1

1 2 1

1 1 2

(a) Find the characteristic polynomial PA(λ) of A and the eigenvalues of A. For convenience, as usual, enumerate the eigenvalues in decreasing order λ1 ≥ λ2 ≥ λ3.

(b) For each eigenvalue λ of A find a basis of the corresponding eigenspace V (λ). Determine (with a motivation) whether V (λ) is a line or a plane through the origin. If some of the spaces V (λ) is a plane find an equation of this plane.

(c) Find a basis of R 3 consisting of eigenvectors if such basis exist. (Explain why or why not). Is the matrix A diagonalizable? If ”yes”, then write down a diagonalizing matrix P, and a diagonal matrix Λ such that A = PΛP −1 , P −1AP = Λ. Explain why the matrix P is invertible but do not compute P −1 .

(d) Consider the eigenvalues λ1 > λ3. Is it true that the orthogonal complements of the eigenspaces satisfy (Vλ1 ) ⊥ = Vλ3 , (Vλ3 ) ⊥ = Vλ1 ? Why or why not??

Let A = 0 2 0

1 0 2

0 1 0 .

(a) Find the eigenvalues of A and bases of the corresponding eigenspaces.

(b) Which of the eigenspaces is a line through the origin? Write down two vectors parallel to this line.

(c) Find a plane W ⊂ R 3 such that for any w ∈ W one has Aw ∈ W , or explain why such a plain does not exist.

(d) Write down explicitly a diagonalizing matrix S, and a diagonal matrix Λ such that S −1AS = Λ; A = SΛS −1 . or explain why A is not diagonalizable.

Letφ:G→G′be a group homomorphism. Prove that Ker(φ) is a normal subgroup of G. Prove both that it is a subgroup AND that it is normal.

Using the table and symbol functions in MSWORD, solve the following proof, making sure to number and justify each line of the proof (including listing premises).

(∀x)(∀y)Pxy ⊢(Ǝx)(Ǝy)Pxy

Question2.

Let A = [2 1 1

1 2 1

1 1 2 ].

(a) Find the characteristic polynomial PA(λ) of A and the eigenvalues of A. For convenience, as usual, enumerate the eigenvalues in decreasing order λ1 ≥ λ2 ≥ λ3.

(b) For each eigenvalue λ of A find a basis of the corresponding eigenspace V (λ). Determine (with a motivation) whether V (λ) is a line or a plane through the origin. If some of the spaces V (λ) is a plane find an equation of this plane.

(c) Find a basis of R 3 consisting of eigenvectors if such basis exist. (Explain why or why not). Is the matrix A diagonalizable? If ”yes”, then write down a diagonalizing matrix P, and a diagonal matrix Λ such that A = PΛP −1 , P −1AP = Λ. Explain why the matrix P is invertible but do not compute P −1 .

(d) Consider the eigenvalues λ1 > λ3. Is it true that the orthogonal complements of the eigenspaces satisfy (Vλ1 ) ⊥ = Vλ3 , (Vλ3 ) ⊥ = Vλ1 ? Why or why not??

***PLEASE SHOW ALL STEPS WITH EXPLANATIONS***

Let G be a group (not necessarily an Abelian group) of order 425. Prove that G must have an element of order 5

A $60,000 bond with a coupon rate of 5.00%, payable semi-annually, is redeemable in 10.5 years. What was the purchase price of the bond, when the yield rate was 5.50% compounded semi-annually?

Block copy, and paste, the argument into the window below, and do a proof to prove that the argument is valid. This question is worth 25 points.

1. (p   • q)   •   (q ⊃   x)
2. ~x v ~s
3. n   v   s     : .    t ⊃ n

Bob makes his first $ 600 deposit into an IRA earning 7.1% compounded annually on his 24th birthday and his last $600 deposit on his 43rd birthday ​(2020 equal deposits in​ all). With no additional​ deposits, the money in the IRA continues to earn 7.1 % interest compounded annually until Bob retires on his 65th birthday. How much is in the IRA when Bob​ retires?

The amount in the IRA when Bob retires is

​(Round to the nearest cent as​ needed.)

Block copy, and paste, the argument into the window below, and do a proof to prove that the argument is valid. This question is worth 25 points.   

1. (p   ⊃   x) • (x ⊃ ~q)
2. p •   n
3.   t v q     : .     t

You're cleaning up your little nephew's toy room. There are T toys on the floor and n empty toy storage boxes. You randomly throw toys into boxes, and when you're done the box with the most toys contains N toys.

(a)What is the smallest that NN could be when T=2n+1?

(b) What is the smallest that NN could be when T=kn+1?

(c)Now suppose that the number of toys T satisfies

T<n(n−1)/2.

Prove that when you are done cleaning there will be (at least) one pair of boxes that contain the same number of toys.