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# Matrices

### Matrices, Vector Spaces and Subspaces

Give a demonstration as to why or why not the given objects are vector subspaces of M22 a) all 2 X 2 matrices with integer entries A vector space is a set that is closed under finite vector addition and scalar multiplication. It is not a vector space, since V is NOT closed under finite scalar multiplication. For insta

### Matrices, Vector Spaces and Zero Vectors

a 1 Show(prove) that the set of all 2 X 2 matrices of the form with addition defined by 1 b a 1 c 1 a+c 1 + = 1 b 1 d

### Size of the Product Matrix

Let A be a 3 × 4 matrix. B be a 4 × 5 matrix, and C be a 4 × 4 matrix. Determine which of the following products are defined and find the size of those that are defined. a) AB b)BA c) AC d) CA e) BC f) CB keywords: multiplying, multiplication, matrices

### Prove the Two Matrices Are Inverse Matrices

Show that [2 3 -1] is the inverse of [7 -8 5 ] [1 2 1] [ -4 5 -3 ] [-1 -1 3] [1 -1 1 ].

### Products of Diagonal Matrices

The n × n matrix A = [aij] is called a diagonal matrix if aij = 0 when i != j. Show that the product of two n × n diagonal matrices is again a diagonal matrix. Give a simple rule for determining this product.

### Addition and Scalar Multiplication of Matrices

Given the set of objects and the operations of addition and scalar multiplication defined in each example below do the following: Determine which sets are vector spaces under the given operations For those sets that fail give the axiom(s) that fail to hold 1. The set of all triples of real numbers(x, y, z) with the op

### Find the eigenvalues of the given 2x2 matrix.

Find the eigenvalues of the following matrix U = mat(2 4; 3 1). See Word document for a clearer version of the problem.

### Finding the inverse of a 2x2 matrix

Find the inverse of the following matrix. Then verify that you have found the inverse. A = mat(4 -3; 8 7). Please see attached Word doc for a cleaner version.

### Determining the Transpose Given the Matrix

Given a 2x3 matrix A, find AA'. A = mat[4 0 7; 9 -3 8]. See attachment for a cleaner version of the problem.

### Find the determinant of 2x2 matrices

Q: Find the determinant of these 2x2 matrices. (a) mat[4 3; -2 9] (b) mat[5 -7; 4 12] Please see the Word document for a cleaner version of the problem.

### Inconsistent Systems and Augmented Matrices

Inconsistent system a. a system with no solution b. a system of inconsistent equations c. a system that is incorrect d. a system that we are not sure how to solve Augmented Matrix a. a matrix with a power booster b. a matrix with no solution c. a square matrix d. a matrix containing the coefficients and constants of

### Graphs : Adjacency Matrix, Order and Valency

Let A be the adjacency matrix of a regular graph of order v and valency k. Let J be the all-ones matrix of the same order. Show that A*J = J*A = K*J "Definition 2.1 A graph r with adjacency matrix A = A(r) is called regular if there exists a natural number k such that AJ = JA = kJ. The number k is called valency of r."

### Linear Binary Codes

You can not use the definition Hermitian inner product to solve this problem. You need to use the definition of weight. We need to generalize the equation. 1-Define the product, x*y , of two binary vectors of the same length to be the vector whose ith component is the product of the ith components of x and y. Show tha

### Binary Hamming Code Vectors

See attached file for full problem description. 7. The parity check matrix of a binary [n, k, d] = [15, 11, 3] Hamming code whose columns are in numeric order is a) Find the corresponding generator matrix. Use the method of permuting and unpermuting the columns for the [7, 4, 3] binary Hamming code. b) Encode the

### Binary repetition code

See attached file for full problem description. 6. Let be the binary repetition code with parameter [n, k] = [12, 4]. a) Give a generator matrix for that is in standard form. What does standard form means? b) Give a parity check matrix for that is in standard form. c) What is the minimum distance of ? d)

### Binary linear block code

See attached file for full problem description. 3. Let be a binary linear block code with parameter [n, k] = [6, 3] which has as its generator matrix. Assume that the first three bits of each codeword are the information bits. a) Find a generator matrix for that is in standard form. What does standard form mea

### Orthogonality of a Generator Matrix of a Binary Block Code

Please see the attached file for the fully formatted problems. 4. Let be the generator matrix of a binary block code . a) Show that each row of G is orthogonal to itself and to each of the other rows of G. b) Show that each codeword in is orthogonal to itself and to every other codeword in . Note: Use part a)

### Binary Hamming code

1-Define the product, , of two binary vectors of the same length to be the vector whose ith component is the product of the ith components of and . Show that wt ( + ) =wt ( ) +wt ( ) - 2wt ( ) . wt means weight. 2. Suppose that a binary Hamming code is modified by adding an additional check bit to each codeword. This ad

### Matrix Decomposition : LU Factorizations, Gram-Schmidt and Gershgorin's Theorem

Please see the attached file for the fully formatted problems. keywords: gershgorin, gershgorins

### Parity check matrix problems

1.- Find a parity check matrix for the [12,4] repetition code. Can you expalin what does mean repetition code and the parity check matrix? 2.- Show that the complement of each codeword in the [12,4] repetition code is again a codeword. Note: do not do this problem by checking all the sums of codewords. Please Can yo

### Solving Matrix Equations Ax=B for x, given A and B

Please see the attached file for the fully formatted problems.

### Matrix Functions

1.Consider the 2 functions f1(t) and f2(t); 1. f1(t) = { a1.e^ -2t for t>=0 and = { 0 for t<0 } f2(t) = { a2.e^ -t + a3.e^ -2t for t>=0 and = { 0 for t<0 } Find a1,a2 and a3 such that f1(t) and f2(t) are orthonormal on the interval 0 to infinity. 2. If Q is an n x n symmetric matrix and a1,a2 are such that 0 < a1I <

### Solving Systems of Equations with Three Variables

1. Production Turley Tailor Inc. makes long-sleeve, short sleeve,and sleeveless blouses. A long-sleeve blouse requires 1.5 hours of cutting and 1.2 hours of sewing. A short-sleeve blouse requires 1 hour of cutting and .9 hour of sewing. A sleeveless blouse requires .5 hour of cutting and .6 hour of sewing. There are 380 hours of

### Matrices Values of Determinants

These are problems from the text that were advised to study for the next exam. (see attachment for equations) 1) Determine the values of r for which det(A-rI) = 0 2) Verify that X(t) is a fundamental matrix for the given system and compute X-1(t). Use the result, x' = Ax, x(t0) = x0 to find the solution to the g

### Determining whether a matrix is invertible

(See attached file for full problem description)

### Vectors, matrices, and polynomials

5) Use cofactor expansion to find the determinant | 4 -1 1 6 | | 0 0 -3 3 | | 4 1 0 14| | 4 1 3 2 | 6)Find the characteristic equation, eigenvalues, and corresponding eigenvectors for A = 1 2 2 1 7) Answer True or False for the following statements: (True means always true; False means sometimes false). Ju

### Vectors and Matrices : Operations and Transformations

1) Let u=(2,3,0), and v=(-1,2,-2). Find a) ||u + v|| b) ||u|| + || v || c) Find two vectors in R³ with norm 1 orthogonal to be both u and v d) Find norm of vector u / || u || 2) For which values of t are vectors u = (6, 5, t), and v = (1,t,t) orthogonal? 3) a) Find the standard matrix [T] for the linear tra

### Trigonometric Applications and Matrices

Trigonometric Applications and Matrices --------------------------------------------- Researchers at the National Interagency Fire Center in Boise, Idaho coordinate many of the firefighting efforts necessary to battle wildfires in the western United States. In an effort to dispatch firefighters for containment, scientists an

### matrix multiplication..

(See attached file for full problem description) 1. Let A = [1 2; 1 4] and B = [ 2 -1; x y]. Are there any x and y such that AB = I? If so, find x and y. 2. Let A = [ 1 0; 1 1] and B = [3 x; 4 y]. Determine the values of x and y such that AB = BA

### Solving Systems of Equations Word Problems and Matrix Inverses

6. How many 7-card hands are possible with a 52-card deck? 7. The owner of nuts2u snack shack mixes cashews worth \$5.50 a pound with peanuts worth \$2.30 a pound to get a half-pound mixed nuts bag worth \$1.80. How much of each kind of nut is included in the mixed bag? 8. Determine whether the matrices are inverse. [5