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# Matrices

### Matrix Questions

1.(a) If A is invertible and AB = AC, prove that B = C. (b) Let A =24 1 1 1 135 explain why A is not invertible. (c) Let A =24 1 1 1 135, ¯nd 2 matrices B and C, B 6= C such that AB = AC. 2. (a) If a square matrix A has the property that row 1 + row 2 = row 3, clearly explain why the matrix A is not invertible. (b)

### Matrix Dimensions and Matrix Division

See attached file for full problem description. Question #2 Only. 2. (a) Find the dimensions of matrix B. Clearly explain your answer. (b) Find matrix B.

### 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

### Matrices

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

### Probability of observable unbounded self-adjoint operator.

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]

### 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&#8712;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 & Vectors: Matrix Products, Ax=B and Linear Combination

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 for which Ax=b does

### Row Operations and Matrix Dimensions

Please see the attached file for the fully formatted problems. 6. Perform the row operation (-2) R1 + R2 &#61664; R2 on the matrix . 8. What are the dimensions of the matrices shown below? a) b) 9. Find

### Matrix Inverses

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

### Invertible Lower Triangular Matrix, Adjoints and Inverses

Find A^-1 using Theorem 2.1.2 - Inverse of a Matrix Using Its Adjoint: If A is an invertible matrix, then A-1 = [1/det(A)] * [adj(A)] A = 2 0 3 0 3 2 -2 0 -4 Prove that if A is an invertible lower triangular matrix, then A-1 is lower triangular.

### 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

### Matrix Symmetry, Matrix Multiplication and Skew-Symmetric Matrices

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

### Solving systems of differential equations by matrix methods.

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

### Differential Equations : Reducing to a first order matrix system

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

### Matrices and Systems of Equations Word Problems

The meteorologists at the National Interagency Fire Center had pizza delivered to their operations center. Their lunch consisted of pizza, milk and gelatin. One slice of cheese pizza contains 290 calories, 15g of protein, 9g of fat, and 39g of carbohydrates. One-half cup of gelatin dessert contains 70 calories, 2g of protein, 0g

### Matrices and Systems of Equations Word Problems

The meteorologists at the National Interagency Fire Center had pizza delivered to their operations center. Their lunch consisted of pizza, milk and gelatin. One slice of cheese pizza contains 290 calories, 15g of protein, 9g of fat, and 39g of carbohydrates. One-half cup of gelatin dessert contains 70 calories, 2g of protein, 0g

### Elementary Matrices, Invertible Matrices and Systems of Equations

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

### Input-Output Matrices, Probability, Matrix Inverses, Probability Distributions and Transition Matrices

1. For the input-output matrix A, and the output matrix X, of three industries, find the amounts consumed internally by the production process. 0.10 0.15 0.10 500 A = 0.20 0.05 0.08 X = 800 0.04 0.06 0.02 400 2. For the input-output matrix A, and the output matrix X

### Hessian Matrix : Maximizing Profit

Question: The profit maximizing input choice A competitive firm's profit function can be written as &#960; := 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

### Laplace Transforms of Matrices

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

### 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?

### Matrix Multiplication and Calculation of Inverses

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

### 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.

### Using Inverse to Find a Multiplying (Multiplier) Matrix

Find the matrix A such that | 1 3 | | 6 5 | A | | = | | | 2 4 | | 1 2 |

### 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