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    Number of Coins Adding Up to a Fixed Amount

    Consider the following problem. Provide evidence for each answer. A collections of k coins consists of nickels, dimes, and quarters, and has a value of 5c cents. The numbers k and c are positive integers. How many coins of each kind are there? (a) Write a system of two equations in three unknowns which models the so

    Sylow-p subgroup

    a. Let G = Gl_3(F_p) denote the multiplicative group of all 3 x 3 nonsingular matrices with entries in the field of p elements F_p = Z/pZ. Show that the set of all matrices of the form (1 a b 0 1 c 0 0 1) where a,b, c belong to F_p, is a Sylow-p subgroup of G. b. Is it the only Sylow-p subgroup of G?

    Describe the graph with a given adjacency matrix. Find the number of simple paths connecting two concrete vertices in this graph. Find the the shortest simple path connecting these vertices. Construct the tree corresponding to the given structure of the college staff. Construct the tree after a certain change in this structure.

    Task Background: Graphs and Trees are useful in visualizing data and the relations within and between data sets. Conversely, it is also important to be able to represent graphs as databases or arrays, so that programs for processing the data can be written. Part I: Adjacency Matrix and Shortest Path Construct a graph based

    standard distance matrix of the TSP model

    An AGV makes a round-trip (starting and ending at the mail room) to deliver mail to 4 departments on a factory floor. Using the mail room as the origin (0, 0), the (x, y) locations of the delivery spots are (10, 30), (10, 50), (30, 10), and (40, 40) for the four departments. All distances are in meters. The AGV moves along horiz

    Cramer's Rules

    The equation of a parabola is: y = Ax2 + Bx + C How can I define the parabola that goes through three points (as we used two points for straight lines)? This solution offers help in developing generic formulas for the parabola that goes through (x1, y1), (x2, y2) and (x3, y3)? It also lists an example situation to a

    Properties of matrices

    Let A=[ -1 2] [ 1 3] a) find A -[1] Use A-1= [ d/ -b/ ] [ ad-bc ad-bc ] [ -c / -a / ] C) Find ( A-[1])3 [ ad-bc ad-bc}

    Examine matrix laws.

    Let A and B be arbitrary n x n matrices whose entries are real numbers. Use basic matrix laws only (donâ??t solve/prove by examples or 2x2 matrices etc.) to expand (A + B )2. Explain all steps. Using basic Matrix laws distributive laws.

    The Eigenvectors of a Self-Adjoint (Hermitian)

    Please see the attachment for the full problem description and hint. Let T >= 0 be a strictly positive definite linear operator on a finite dimensional inner product space V over F = R or C. (a) Prove that the exponential map Exp: A -> e^A = sum infinity k = 0 1/kl A^k is one-to-one from the space of self-adjoint operators

    set of all invertible diagonal matrices

    Let R be any ring with identity 1 and GL(n, r) the group of invertible nxn matrices over R and SL(n, R) those matrices of determinant 1. 1) Show that SL(n, R) forms a normal subgroup of GL(n, R) 2) Define a group of homomorphism fronm GL(n, R) to another group for which SL(n, R) is the kernel 3) Determine the center of the gr

    A 2 by 2 rotation matrix

    Question 4: Give an example of a 2 by 2 rotation matrix that rotates the point ( 4 , 5 ) into the square with corners ( -3 , -7 ) ( -1 , -7 ) ( -1 , -5 ) ( -3 , -5 ) You will probably need to experiment to find a suitable rotation. You can c

    Matrix multiplication

    1 -1 4 3 4 2 2 - i 3 A:=3 1 0 B:=5 1 a:=0 c:= 4 + i d:= 1 + 5i 1 2 1 0 -3 1 5 - i 2 - i matrix multiplication: 1) AxB 2) Axc

    Applying the Gauss-Jordan Method

    Use the Gauss-Jordan method to solve the system of equations. −3x + 4y = 17 x − 5y = −13 Enter the answer as a coordinate pair including the parentheses and comma. If a coordinate is not an integer, enter it as a fraction in simplest form. If the system has no solution, "no solution" should be entered.

    Example Questions: Matrix Equations

    2x + y = 6 x - y = -3 Write down A matrix using left side of each equation. Find inverse matrix A -1. Multiply matrix A -1 by column of numbers on right side of the equations. Result will be a column of two numbers: first - the answer for x, second the answer for y.

    Multiplying matrices

    Matrices Multiply: [■(1&2@3&0)][■(1@-1)] = Multiply: [■(-1&2@3&1)][■(2&4@3&1)] = Matrix A shows number of stocks holding by William and Michael: Matrix B shows price in dollars per share: Find inverse to the given matrix (see instructions on Blackboard in Course

    Product and inverse of matrices

    Assign any numbers for A-matrix 2x2 and B-matrix 2x2. A B | ... ... | | ... ... | | ... ... | | ... ... | Find matrix that is product of multiplication A x B. and find inverse matrix A -1

    Expected market share

    Please help with the following problem. Provide step by step calculations. Consumers in a certain state can shoose between three long distance telephone services: GTT, NCJ and Dash. Aggressive marketing by all three companies results in continual shift of customers among the three services. Each year, GTT loses 30% of its cu

    Study transformations represented by matrices.

    Please refer to attachment for proper formatting. (i) Write down the matrix, A, that represents a shear with x-axis invariant in which the image of the point (1,1) is (4,1). (ii) Given matrix B (in the attachment), describe fully the geometrical transformation represented by B. (iii) Given matrix C (in the attachment),

    conjugation by elementary matrices

    (i) Let G = SL2(R). Using conjugation by elementary matrices, show that every matrix A in G except for +-I is conjugate to a matrix having one of the forms | 0 -1 | | 1 d | or | 0 1 | |-1 d |. (ii) Let A = | x y | | z w | be a matrix in G and let t be its trace. Substituting t-x for w, the condi

    Positive definite matrix

    Which 3 by 3 symmetric matrices A produce following functions f = X^T A X ? Why is the first matrix positive definite but not the second one? a) f = 2 (X1^2 + X2^2 + X3^2 - X1X2 - X2X3) b) f = 2 (X1^2 + X2^2 + X3^2 - X1X2 - X1X3 - X2X3)

    Input-output Closed model.

    1) Use the input-output matrix and the Closed Model to find the ratio of yams to pigs produced. 2) Find [I-A]. Solve [I-A] [X]=0 Yams Pigs Yams 1/4 1/2 Pigs 3/4 1/2

    orthogonal matrix group

    How can I show that O(n) and SO(n) x Z2 are homeomorphic? O(n) is the orthogonal group and SO(n) is the special orthogonal group.

    Use elementary row operations

    Solve the following systems. Use elementary row operations to reduce the augmented matrix of the system to our upper diagonal matrix. a) [x+y+z=2] [2x-y-z=1] [3x+2y+z=3] b) [x+y+z=2] [2x-y-z=1] [4x+y+z=3]

    Finding the Inverses of Two Matrices

    Hey, I am having trouble computing the inverse of a matrix here. I can get close to it, but I am unsure how to go forward with it. 1) Find A^-1 for A= [1 0 1] [2 1 0] [3 2 2] 2) Find the inverse of A = [2 -3]

    available production capacity

    Freds furniture factory has 1950 machine hours available eack week in the cutting department, 1490 hours in the assembly department , and 2160 in the finishing department. manufacturing a chair requires 0.2 hours of cutting, 0.3 hours of assembly, and 0.1 hours of finishing. a cabinet requires 0.5 hours of cutting, 0.4 hour

    Compute an invertible matrix

    Consider the Matrix A given in the attachment. By considering the appropriate row operations and their associated elementary matrices, compute an invertible matrix B (meeting given criteria) such that B.A = [[1 0 * *] [ 0 1 * *] [ 0 0 * *]], where * denotes a nonzero number. Please see the attachment for complete question.

    Algebra: Matrix Operations

    1) At a store, Sam bought 3 batteries, 15 60-watt light bulbs, 46 100-watt light bulbs, 8 picture-hanging kits, and a hammer. Jennifer bought 12 batteries, 3 100-watt light bulbs, and a package of tacks. Write the information as a 2 x 6 matrix. See the attached file.

    Formulate a matrix equation and solve it.

    Make sure to show all of the steps you used. a) Set up the matrix equation to solve the problem given below. Transportation Ace Trucking Company has an order for three products, A, B, and C, for delivery. The table below gives the volume in cubic feet, the weight in pounds, and the value for insurance in dollars for a uni