Let 1, 2, ..., n be a complete set of eigenvalues (repetitions included) of the n

Question:

Let λ1, λ2, ..., λn be a complete set of eigenvalues (repetitions included) of the n × n matrix A. Prove that
det(A) = λ1 λ2 ··· An and
tr(A) = λ1 + λ2 + ··· + λn

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Linear Algebra A Modern Introduction

ISBN: 978-1285463247

4th edition

Authors: David Poole

See More Books

Question Posted: Apr 15, 2016 06:14 AM

See More Questions

Let A be a square matrix that can be partitioned as where P and S are square matrices. Such a matrix is said to be in block (upper) triangular form. Prove that det A = (det P)(det S) o S
Let M > 0 be a fixed positive definite n n matrix. A nonzero vector v 0 is called a generalized eigenvector of the n n matrix K if Kv = Mv, v 0, (8.31) where the scalar is the corresponding...
(a) Find the explicit formula for the characteristic polynomial det(A - I) = -3 + a2 - b + c of a general 3 3 matrix. Verify that a = tr A, c = det A. What is the formula for b? (b) Prove that if A...
Use (4.1), for a perfectly conducting fluid, and the nonlinear equation of continuity (1.1), to show that the change of B with time in a fluid element is related to changes of density according to...
Foyle Hardware reported cost of goods sold as follows. Foyle made two errors: 1. 2013 ending inventory was overstated by $2,000. 2. 2014 ending inventory was understated by $5,000. Instructions...
Consider the normal shock wave properties calculated in Example 3.5. Show that these properties satisfy the Hugoniot equation for a calorically perfect gas. Data From Example 3.5: A normal shock wave...
Consider two countries, A and B, with the technologies given by case 3 in Exercise 1. data from case 3 in exercise 1 Suppose that the wage rate in A, WA, equals $$9$ per hour; then, for mutually...
Atmospheric air enters the heated section of a circular tube at a flow rate of 0.005 kg/s and a temperature of 20C. The tube is of diameter D = 50 mm, and full developed conditions with h = 25 W/m 2 ...
4. (20 points) Minimize the following DFA using either Brzozowski's algorithm or Hopcroft's algorithm, show your steps. b a a 92 a a 94 b b 14 a b b 46 a
Three mutually exclusive projects are being considered for a remote river valley: Project R, a recreational facility, has estimated benefits of $20 million and costs of $16 million; project F, a...
Show that if the square matrix A can be partitioned as where P and S are square matrices, then the characteristic polynomial of A is cA(Î») = cp(Î») cS(Î»). Q Lo s
Let A and B be n n matrices. Prove that the sum of all the eigenvalues of A + B is the sum of all the eigenvalues of A and B individually. Prove that the product of all the eigenvalues of AB is the...
Determine the steady-state surface temperature of an electric cable, 25 cm in diameter, which is suspended horizontally in still air in which heat is dissipated by the cable at a rate of 27 W per...
What are some of the pros and cons of theft law consolidation?
Defendants Elie and Campos both are charged with three counts of violation of the Unlawful Internet Gambling Enforcement Act of 2006 (UIGEA), one count of conspiracy to violate UIGEA, two counts of...
For purposes of defining homicide, when does life begin and end? Why?
Should gambling be illegal? What about social gambling? Defend your answer.
What is the difference between first-degree and second-degree murder?
In Figure 10-8 we showed the value of Chinese exports of rare earth minerals, along with their average price and quantity sold, in three categories of exports. The source for the data in Figure 10-8...
Ann hires a nanny to watch her two children while she works at a local hospital. She pays the 19 year-old nanny $125 per week for 48 weeks during the current year. a. What is the employers portion of...
Compute the indicated vectors. 2a + 3c Data From Exercise 3 (a) a = [0, 2, 0] (b) b = [3, 2, 1] (c) c = [1, -2, 1] (d) d = [-1, -1, -2]
Compute the indicated vectors. 2c - 3b - d Data From Exercise 3 (a) a = [0, 2, 0] (b) b = [3, 2, 1] (c) c = [1, -2, 1] (d) d = [-1, -1, -2]
Draw the coordinate axes relative to u and v and locate w. w = -u - 2v 3 ||
On-campus corporate presentations: Sympathy for the Devil Many students complain about the high number of corporate presentations taking place both on- and off-campus. In this question, we seek to...
How do organizations cultivate a culture of open innovation, leveraging crowdsourcing platforms, collaborative networks, and cross-industry partnerships to harness external knowledge and drive...
Braeden Sim is the accountant for Sim's Internet Service. From the following information, his task is to construct a balance sheet as of April 30, 201X, in proper form. Could you help him? (Click the...

Let 1, 2, ..., n be a complete set of eigenvalues (repetitions included) of the n

Question:

Step by Step Answer:

Start with the equation given in the problem statement This is an equation so in particular it holds ...View the full answer

Linear Algebra A Modern Introduction

Students also viewed these Linear Algebra questions