Prove that if A and B are n n matrices, then (a) tr(KA) = k tr(A)

Question:

Prove that if A and B are n × n matrices, then
(a) tr(KA) = k tr(A) + tr(B)
(b) tr(AB) = tr(BA)

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

Step by Step Answer:

Related Book For book-img-for-question

Elementary Linear Algebra with Applications

ISBN: 978-0471669593

9th edition

Authors: Howard Anton, Chris Rorres

See More Books

Question Posted: Jun 17, 2016 07:58 AM

See More Questions

(a) Give an example to show that if A and B are symmetric n n matrices, then AB need not be symmetric. (b) Prove that if A and B are symmetric n n matrices, then AB is symmetric if and only if AB =...
Prove that if A and B are skew-symmetric n n matrices, then so is A + B.
Upper Triangular Matrices A square n ( n matrix A = [aij] is called upper triangular if aij = 0 for i > j. (All entries below the main diagonal are zero.) (a) Give three examples of upper triangular...
The emergency room of the community hospital in Farmburg has one receptionist, one doctor, and one nurse. The emergency room opens at time zero, and patients begin to arrive some time later. Patients...
Lynn Price recently completed her MBA and accepted a job with an electronics manufacturing company. Although she likes her job, she also is looking forward to retiring one day. To ensure that her...
The professional association CPA Canada requires its members to maintain a commitment to: a. Environmental sustainability reporting b. Ethical decision making to benefit the largest number of...
Discuss the importance of the patients responsibility to communicate to the staff any symptoms he or she is experiencing.
A plastic ball has radius 12.0 cm and floats in water with 16.0% of its volume submerged. (a) What force must you apply to the ball to hold it at rest totally below the surface of the water? (b) If...
If I threw a 1 . 0 kg text book up into the air with a speed of 5 . 0 km / s , what maximum height would it reach? The mass of the Earth is 5 . 9 8 x 1 0 ^ 2 4 kg and its radius is 6 . 3 8 1 0 ^ 6...
F. Pierce Products Inc. is considering changing its capital structure. F. Pierce currently has no debt and no preferred stock, but it would like to add some debt to take advantage of low interest...
Prove: If B is invertible, then AB-1 = B-1 A if and only if AB - BA.
Use part (c) of Exercise 24 to show that State all the assumptions you make in obtaining this formula. dA- - A-1dA A -1 dz dx
Ethylenediaminetetraacetic acid displaces bismuth( III) from its thiourea complex: Bi(tu)63+ + H2Y2- ( BiY- + 6tu + 2H+ Where tu is the thiourea molecule, (NH2)2CS. Predict the shape of a photometric...
Before boarding her flight to Zurich, Switzerland, Mary purchased CHF850 from her bank when the exchange rate was C$1 = CHF0.9651. However, Mary had to cancel the trip. Mary returned to the bank to...
Robert purchased US$15,000 from a bank in America, which charged him a commission of 0.9%, and sold the US dollars to a bank in Canada, which charged him a 0.24% commission. How much money did he...
Consider the following set for n N: F = {f : {1, . . . , n} {0, 1, 2} :1 i n 1, f(i) + f(i + 1)4}. (a) Given a recurrence and boundary conditions for |F|. (b) Solve the recurrence and give an...
1. Brody knows that he can afford a monthly mortgage payment of $1500. The bank will offer him a mortgage rate of 2.14% compounded monthly over a 25 years period. Calculate the most expensive home...
Maya purchased a boat for $18,340. Its value depreciated by 15% in the first year she owned it. What was her boat worth at the end of this first year?
In Problem find the vertex form for each quadratic function. Then find each of the following: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range u(x) = 0.5x 2 - 2x + 5
Rowland Textile Inc. manufactures two products: sweatshirts and T-shirts. The manufacturing process involves two activities: cutting and sewing. Expected overhead costs and cost drivers are as...
Let V be an inner product space. If u and v are vectors in V, Show that (u, v) = ||u + v||2 -1/4 ||u-v||2
Let V be an inner product space and u a fixed vector in V. Prove that the set of all vectors in V that are orthogonal to u is a subspace of V.
If V is an inner product space, prove that the distance function of Definition 5, 3 satisfies the following properties for all vectors u, v, and w in V: (a) d(u. v) 0 (b) d(u, v) = 0 if and only if...
Work out which number line below shows the values that x can take if 3 x 2. - A +12345678910 8 9 10 B D 5 6 7 8 9 10 6 7 8 E 2 F 3 4 7 8 9 10 1 2 3 4 5 6 7 8 9 10
Application Case Coffee Shop is a company dedicated to the commercialization of "signature coffee", a category of high quality coffee that is grown, processed and prepared with a high degree of...
The issuing company Zapatos y Bolsas Veloz, SAB de C.V., dedicated to the manufacture and sale of vinyl leather bags for more than 25 years under the Bolsas Finas brand, the company obtained in the...

Prove that if A and B are n n matrices, then (a) tr(KA) = k tr(A)

Question:

Step by Step Answer:

a Let the ij th entry of A be a ij Then tr A a 11 a 22 a nn so that t...View the full answer

Elementary Linear Algebra with Applications

Students also viewed these Linear Algebra questions