Explore BrainMass
Share

# Matrices

### Hubert Matrices Using Matlab

C) The famous Hubert matrices are given by Hij= 1/(i +j - 1). The n x n Hilbert matrix Hn is easily produced in MATLAB using hilb(n). Assume the true solution of H,x = b for a given n is x = [1.. .. , 1]^T. Hence the righthand side b is simply the row sums of H, and b is easily computed in MATLAB using b=sum(hilb(n) 9'. Use your

### Gaussian Elimination and Operation Count

Consider the problem Ax=b where A is a tridiagonal matrix. What is the operation count for the forward elimination and the back substitution steps of Gauss elimination in this case? Count add/sub and mult/div operations separately, then give the overall order of the total operations needed. (Use O(n^p) notation).

### Products of a Matrix and its Transpose

Find X X' and X' X where X =[ 1 -1 2]. Note: X' is the transpose of X. See the attached file.

### Vectors, Matrix Operations and Components

10. Let A be an M x N matrix and X an N x 1 matrix. (a) How many multiplications are needed to calculate AX? (b) How many additions are needed to calculate AX?

### Real symmetric matrices and the Cauchy-Schwarz inequality.

Let A be a real square symmetric matrix and let S(A) denote the sum of all entries of A, show that S(A) / S(I) < S(A^2) / S(A)

### Symplectic Matrix, Determinant, Characteristic Polynomial and Eigenvalues

If ' is a symplectic matrix show that det ' = 1. Also prove that....where P(...) = det(,,,) is the characteristic polynomial of '. Finally, if ... is an eigenvalue with multiplicity k of ' then the same is true for...

### Determinants, Transposes and Row Reduction

(a b c) (k 2(a-k) p+k) Given det (k l m) = d find det ( l 2(b-l) q+l) (p q r) (m 2(c-m) r+m) Please see the attached file for the fully formatted problem.

### Find the Determinant and Inverse of a Matrix

(1) Let A = (see attachment), the nxn matrix with all entries equal to 1 expect diagonal entries, which are equal to 0. Find the determinant . (2) For the above matrix A, find the inverse. Please see attachment for complete questions.

### Matrix Equations : Inverse and Determinant

The solution X for the matrix equation X-AX=D is: _____

### 1. This question is concerned with subgroups of the group S5 of symmetries (or permutations) on the set {1,2,3,4,5}, a group with 120 elements. (a) Explain why this group has cyclic subgroups of order 1,2,3,4,5 and 6 and give examples of each of these. Explain why this group does not have cyclic subgroups of any order. (b) By considering the symmetry groups of appropriate geometric figures, give examples of : (iv) a subgroup of order 4 that is not cyclic; (v) a subgroup of order 6 that is not cyclic; (vi) a subgroup of order 8. (c) By considering those permutations that fix one element, or, otherwise, give an example of a subgroup of order 24 and another of order 12. [You need not list all the elements of these groups, but you should explain clearly which elements constitute each subgroup.] (d) List the potential orders of subgroups of S5 (other than S5 itself), according to Lagrange's theorem, in addition to those already considered in this question. Give an example of a subgroup of one of these orders. 2. (a) Which of the following sets are groups under the specified binary operation? In each case, justify your answer. (ix) Z, the set of integers, under operations * defined by a*b = a + 2b (x) R*, the set of non-zero real numbers, under the operation × defined by x×y = 5xy (xi) The set {3,6,9,12} under multiplication modulo 15. (xii) The set of matrices {(1, p;0,1)/p&#1028;Z} under matrix multiplication. (b) G is a group of real functions with domain and co-domain the non-negative real numbers, i.e. functions [0,&#8734;) &#8594; [0,&#8734;). The group operation in G is function composition. If one of the elements of G is the squaring function, f, defined by f(x) = x2 , explain why G must be an infinite group. 4. (a) Define the motion of conjugacy as it applies in a general group. Prove that the inverses of a pair of conjugate elements are also conjugate. Prove that conjugate elements have the same order. The remainder of this question concerns the group G, whose Cayley table is as follows: e a b c d f g h i j k l ----|-------------------------------------------------------------------------------------------------- e | e a b c d f g h i j k l a | a b e d f c h i g k l j b | b e a f c d i g h l j k c | c f d e b a j l k g i h d | d c f a e b k j l h g i f | f d c b a e l k j i h g g | g h i j k l e a b c d f h | h i g k l j a b e d f c i | i g h l j k b e a f c d j | j l k g i h c f d e b a k | k j l h g i d c f a e b l | l k j i h g f d c b a e (b) Determine the inverse and the order of each of the elements of G. (c) Simplify each of the following: (i) acb (ii) bca (iii) ajb (iv) bja (v) gcg (d) Given that the only element conjugate to g is g itself ( you need not prove this), determine the conjugacy classes of G. (e) Find H, a normal subgroup of G having three elements. Identify the elements of the quotient group G/H and determine its isomorphism type.

Modern Algebra Group theory Symmetric Groups Permu

### Associative & Commutative Rule

Prove that addition modulo n, written +n is: 1) associative 2)comutative there are two ways to prove these properties. each way requires a definition or two: 1) for n&#8805;2, 0&#8804;a, b&#8804;n+1 a+n(written as a power in a corner downside, but dont know how to put it tho) b={condition 1 - a+b if a+b<n;condition 2 - a

### Determinants, Cofactors and Permutations

Q1. Suppose An is the n by n tridiagonal matrix with 1's everywhere on the three diagonals... Let Dn be the determinant of An; we want to find it. (a) Expand in cofactors along the first row of An to show that Dn = Dn-1 - Dn-2 (b) Starting from D1 = 1 and D2 = 0 find D3, D4, ..., D8. By noticing how these numbers cycle a

### Properties of Determinants: Orthogonal Matrix and Parallelepiped

If Q is an orthogonal matrix, so that QTQ = I, prove that det Q equals +1 or -1. What kind of parallelepiped is formed from the rows (or columns) of an orthogonal matrix Q? See the attached file.

### Gaussian Elimination and Back Substitution

X + 2y + z = 0 -3x + 3y + 2z = -7 4x - 2y - 3z = 2

### Matrix Operations Inverse Functions

Given A = 1 3 -2 6 Find A^-1, the inverse of A Please show ALL work, thank you!

### Matrix Operations : Transpose

Given A = 1 3 2 Find A^T

### Matrix Operations : Multiplication of a Matrix by a Constant

Given r = -3 A = 1 3 2 -1 Find rA

### Matrix Operations : Matrix Addition

Given A = 1 2 -2 3 4 5 B = 2 0 1 3 -2 5 C = -4 -6 1 2 3 0 a.) Find A+B and B+A b.) Find A+B+C

### Systems of Equations : Real World Situations and Determinants

1. In real-world situations, what is the advantage of using the Method of Substitution to solve a system of equations rather than using the Method of Addition? 2. When solving a 3x3 determinant, we broke the determinant down into a sequence of 2x2 determinants, remembering to alternate the signs of the leading coefficients in

### Equations, point of intesection, plane of intersection

14) Solve the following system of equations x1 - 3 x2 =5 -x1 + x2 + 5 x3= 2 x2 + x3 =0 15) Determine if the system is consistent. Do not completely solve x1 + 3 x3 =2 x2-3x4 =3 -2x2+ 3x3 + 2 x4= 1 7 x4= -5 17) Do the three lines x1 -4 x2 = 1, 2 x1 -x2 = -3, - x1 -3 x2 =4 , have a common point of intersection? Exp

### Matrices

Need to see problems on attachment done to further grasp concepts I am missing. Please do not leave out any details in solving the problems. If I ask questions please answer them in detail and please make everything simple and clear to understand. My goal is to apply what you solve to other problems I want to work on on my ow

### Matricies

Need to see problems on attachment done to further grasp concepts I am missing. Please do not leave out any details in solving the problems. If I ask questions please answer them in detail and please make everything simple and clear to understand. My goal is to apply what you solve to other problems I want to work on on my ow

### Matrices : Solve by Row Reduction

Please show every step no matter how minor, use the brackets for each reduction and write out every equation change. Please leave no details out. Thanks! 1) Find the general solutions of the systems whose augmented matricies are given in a) and b). What is a general solution? a)1 -2 -1 3 3 -6 -2 2 b) 1 2 -5 -

### Matrices : Gauss-Jordan Elimination

65. A father, when dying, gave to his sons 30 barrels, of which 10 were full of wine, 10 were half full, and the last 10 were empty. Divide the wine and flasks so that there will be equal division among the three sons of both wine and barrels. Find all the solutions of the problem. (from Alcuin) 4, 5 Find all solutions of the

### Matrices: Gaussian Elimination, Calculation Time and Cramer's Rule

Questions: a) How many multiplications are necessary to find the determinants of matrices which are 2x2, 3x3, 4x4? b) The number of multiplications for an nxn matrix may be found in terms of the number for an (n-1)x(n-1) matrix. FIND THIS FORMULA and use it to obtain the number of multiplications for a 10x10 matrix. c) Fo

### Matrices : Number of Terms

The number of multiplications for an n X n matrix may be found in terms of the number for an (n-1) X (n-1) matrix. Find this formula and use it to obtain the number of multiplications for a 10 X 10 matrix

### Matrices : Gaussian Elimination and Analysis of Calculation Time

How many multiplications are necessary to find the determinants of matrices which are (2,2) (3,3) and (4,4)? The number of multiplications for an n,n matrix may be found in terms of the number for (n-1) X (n-1) matrix. Find this formula and use it to obtain the number of multiplications for a 10,10 matrix. For an nXn matri

### Null Space: What is the Range?

Show that the null space of A^A coincide with the null space of A. What is the range? See the attached file.

### Matrix System explained

Solve the system attached . Give your solution in real form. Solve the system -3 -3 3 -3 with 1 -1

### Matrix System and Real Form

Solve the system attached. Give your solution in real form. -9 3 -30 9 with 2 1