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Invertible Matrix over Complex Numbers

Let A be a square n x n matrix over C[X] and write A = [pjk (X)] . For any z∈C( z being a complex variable) let A(z) := [pj k (z)] , that is a square n x n matrix over C. Show that matrix A is invertible if and only if matrix A(z) is invertible for all z from C. Will it be still valid if we change complex numbers into

Matrices and Pivot Positions : Solutions for Ax+0 and Ax=b

In problems 1-4 (a) does the equation Ax = 0 have a nontrivial solution and (b) does the equation the Ax = b have at least one solution for every possible b? 1) A is a 3x3 matrix with three pivot positions. 2) A is a 3x3 matrix with two pivot positions. 3) A is a 3x2 matrix with two pivot positions. 4) A is a 2x4 matrix

Matrices And Vectors : Matrix Products, Ax=B and Linear Combinations

Please see the attached file for the fully formatted problems. Compute the products using the row vector rule for computing Ax. If a product is undefined, explain why. 1) 2) 3) 4) let A = and b = . Show that the equation Ax=b does not have a solution for all possible b, and describe the set of all b f

Row Operations and Matrix Dimensions

Please see the attached file for the fully formatted problems. 6. Perform the row operation (-2) R1 + R2  R2 on the matrix . 8. What are the dimensions of the matrices shown below? a) b) 9. Find

Proof : Sum of Matrix Columns

If the sum of any of the columns of a matrix is 1 and that of any row is 1 then prove that there are equal number of rows and columns.

Lesson 5631-2: Matrices

43 Matrix Problems. See attached file for full problem description. 1. The symobl [A] denotes a 2. For a mn matrix (m rows and n columns) when m = n, the matrix is said to be 3. The matrix [0 2 3] is a 4. The number of columns in a column matrix is 5. In the matrix [A] = [ 1 6; 5 2; 0 -3], the element a_32 is

Hessian Matrix : Maximizing Profit

Question: The profit maximizing input choice A competitive firm's profit function can be written as π := p * q - w * L - r * k where p is the competitive price of the product and w and r the per unit cost of the two inputs labour (L) and capital (k). the firm takes p,w, and r as given and chooses L and k to maximize

Coding Theory : Vectors and Generator Matrices

Please see the attached file for the fully formatted problems. 1(i) Explain what is meant by (a) a linear code over Fq, (b) the weight w(u) of a vector u and the distance d(u, v) between vectors u and v. (c) Define the weight tu(C) and the minimal distance d(C) of a code C. Prove that w(C) = d(C) if C is linear. (ii) Give

Groups and Matrices

1 3 1 4 Let the matrix A = 0 2 and the matrix B = 5 1 be elements in GL(2, Z_7). Find (A^-1 * B^-1)^-1. - I am unsure of when to perform the operation mod 7.

Laplace Transforms of a Matrix System

Use the Laplace transform approach to find to find y(t) for the system given by Please see the attached file for the fully formatted problems. keywords: matrices, transformations

Direct Products and Isomorphisms

Let G = Z_3 direct product Z_3 direct product Z_3 and let H be the subgroup of SL(3, Z_3) consisting of 1 a b the matrix H = { 0 1 c with a, b, c in Z_3 } 0 0 1 What is the order of G and H and are G and H isomorphic?

LaPlace Transforms for Circuit Schematics

Write the LaPlace-transformed loop equations for these two circuits by inspection. Use matrix notation. Include initial conditions. See attached file for full problem description.

Linear Combinations of Matrices

4 0 1 -1 0 2 A = B = C = -2 -2 2 3 1 4 Which of the following 2 X 2 matrices are linear combinations of A, B, or C 6 -8 0 0 6 0 -1 5

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

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

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 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.

High School Algebra Questions

4. dependent system a) a system that is dependent b. a system that depends on a variable c. a system that has no solution d. a system for which the graphs coincide 8. Matrix a) a movie b) a maze c) a rectangular array of numbers d) coordinates in four dimensions 12. Sign array a) the signs of the entries of