### Scalar Multiplication of Matrices

For what value of k does equality hold l 5 2 3 l l 1 2 3 l and l 1 2 4 l l 1 2 4 l l -10 3 4 l =k l -2 3 4 l ? l 3 6 9 l =k l 3 6 9 l l-15 4 5 l l -3 4 5 l l 0 5 0 l

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For what value of k does equality hold l 5 2 3 l l 1 2 3 l and l 1 2 4 l l 1 2 4 l l -10 3 4 l =k l -2 3 4 l ? l 3 6 9 l =k l 3 6 9 l l-15 4 5 l l -3 4 5 l l 0 5 0 l

Find the two numbers whose sum is 76 and quotient is 18. Evaluate the determinate l 8 0 0 l l -16 7 8 l l 8 4 5 l FInd A-1(power) where A = l 2 4 l l 2 5 l

Let H be the unbounded self-adjoint operator defined by -d^2/dx^2 (the negative of the second derivative with respect to x) on: D(H) ={f element of L^2 | Integral( |s^2 F f(s)|^2 )ds element of L^2} Where "F" denotes the Fourier Transform. Question: For the state vector h(x) = 1/sqrt(2) if x is in [0,2]

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

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

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

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

Find A-1 where A= [2 4] [2 5]

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.

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

2. Compute the product by inspection. a) 3 0 0 2 1 b) 2 0 0 4 -1 3 -3 0 0 0 -1 0 -4 1 0 -1 0 1 2 0 0 5 0 0 0 2 2 5 0 0 4 -5 1 -2 0 0 2 8. Use the given equ

Solve each of the following systems by matrix methods. #2) x''' = 6t when x(0)=0, x'(0)=0, x''(0)=12

Reduce each of the following systems to a first-order matrix system: #21.10) x'' - 2x' + x = t + 1 when x(1)=1 and x'(1)=2 #21.13) y + 5y' - 2ty = t^2 + 1 when y(0)=11 and y'(0)=12 keywords: IVP, initial value problems, ODE

1. Consider the matrices A = 3 4 1 B = 8 1 5 C = 3 4 1 2 -7 -1 2 -7 -1 2 -7 -1 8 1 5 3 4 1 2 -7 3 Is it possible to find an elementary matrix E such th

Questions 5 - 10 . Please show step by step if possible and the answers. Please see the attached file for the fully formatted problems.

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

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

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.

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

Use the Laplace transform method to find e^At given for A. A = [-1 0 ] [ 0 4] keywords: matrix

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?

Using A = [ cos a -sin a sin a cosa ] Find A inverse Check A is in So sub 2 (R) Check A inverse *A = Identity and A* A inverse = Identity Show that S) sub 2 (R) is abelian

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.

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

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

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

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

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

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.