Show the values in A and b after each row operation and then find the values
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It seems that the answer to question 7 is missing. May I ask for the answer and explanation for question 7?
![Question 7 (Gaussian elimination) DI SAAF DI SZAG DI KAAG DI KAG DI KA
Apply forward elimination to this system of linear equations to put it into the upper triangular form.
2
6300
2 2
2 5
=
13
2000:
Write each row operation you use to solve the problem.
Find the values of x, y, and z using back substitution.
Write the matrix and the right-hand side after each row operation, for example, the initial matrix and
right-hand side can be represented as:
04
A = [[1, 2, -1], _ _ _ _§Ù ÙƒÙ ÙƒÙ 0¾0 0¾Ò 0;
[1, 2, 2],
[2, 5, -1]]
b = [6, 3,13]
Note: Do NOT write code. Show the values in A and b after each row operation and then find the values
of x, y and z using back substitution.G LIKE](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F508a77ab-0a17-47f4-81a6-47a24375402a%2Fc00bfec3-44cf-4189-8008-c8e83762a5a1%2Ftkbyupi_processed.png&w=3840&q=75)
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- Solving system of linear equations with two variables using Cramer’s rule. In mathematics the coefficientsof linear equations are represented in the form of matrices. A matrix is a two-dimensional array inprogramming. The steps of Cramer’s rule to solve the system of equations are:Step 1: Determining the coefficient matrix from linear equations e.g. a1x+b1y=d1a2x+b2y=d2Coefficient matrix: D = Step 2: Determining the constant column C = Step 3: Finding determinant |D| = (a1*b2) - (b1*a2) and checking if |D| is not equal to zero then do thefollowing: 1. Determining X-matrix: DX = The coefficients of the x−column are replaced by the constant column a1 b1a2 b2d1d2 d1 b1d2 b2 2. Determining Y-matrix: Dy = The coefficients of the Y−column are replaced by the constant column3. To solve for x: x=|DX|/|D|4. To solve for y: y=|Dy|/|D|Note that if |D|=0 then the matrix is singular and solving the system of equationis not possible. You’re required to automate all these steps by writing a program…Linear Equation SystemLU Decomposition Method AlgorithmDecomposition phase The LU Decomposition (Doolittle) method has the following properties: The U matrix is identical to the upper triangular matrix resulting from the Gaussian Elimination;The elements at the bottom below the main diagonal of the matrix L are the multipliers used during the Gaussian Elimination, that is, Li*j is the multiplier that eliminated Ai*j.It is common practice to store the multipliers at the bottom of the coefficient matrix, replacing the coefficients as they are eliminated (Li*j replacing Ai*j) The diagonal elements of L do not need to be stored, as their values are understood to be unitary. The final form of the coefficient matrix would be the mixture of L and U: The initial part of the LU Decomposition Method algorithm is identified with Gaussian Elimination, except that each λ multiplier is now stored in the lower triangular portion of matrix A: solve the system of linear equations followed by the…python Linear Equation SystemLU Decomposition Method AlgorithmDecomposition phase The LU Decomposition (Doolittle) method has the following properties: The U matrix is identical to the upper triangular matrix resulting from the Gaussian Elimination;The elements at the bottom below the main diagonal of the matrix L are the multipliers used during the Gaussian Elimination, that is, Li*j is the multiplier that eliminated Ai*j.It is common practice to store the multipliers at the bottom of the coefficient matrix, replacing the coefficients as they are eliminated (Li*j replacing Ai*j) The diagonal elements of L do not need to be stored, as their values are understood to be unitary. The final form of the coefficient matrix would be the mixture of L and U: The initial part of the LU Decomposition Method algorithm is identified with Gaussian Elimination, except that each λ multiplier is now stored in the lower triangular portion of matrix A: 1)solve the system of linear equations…
- numerical analysis question/Answer according to the system of linear equations given in the picture: Write it in the form of a matrix with [A ⋮ B] additions. Solve the system of linear equations with the Gaussian elimination method by applying elementary row operations to the [A ⋮ B] addition matrix and showing them one by one at each step.(Sparse matrix–vector product) Recall from Section 3.4.2 that a matrix is said to be sparse if most of its entries are zero. More formally, assume a m × n matrix A has sparsity coefficient γ(A) ≪ 1, where γ(A) ≐ d(A)/s(A), d(A) is the number of nonzero elements in A, and s(A) is the size of A (in this case, s(A) = mn). 1. Evaluate the number of operations (multiplications and additions) that are required to form the matrix– vector product Ax, for any given vector x ∈ Rn and generic, non-sparse A. Show that this number is reduced by a factor γ(A), if A is sparse. 2. Now assume that A is not sparse, but is a rank-one modification of a sparse matrix. That is, A is of the form à + uv⊤, where à ∈ Rm,n is sparse, and u ∈ Rm, v ∈ Rm are given. Devise a method to compute the matrix–vector product Ax that exploits sparsity.Replace the first row of matrix Y to vector b , b=[4 6 8 10]
- Q1) Gram-Schmidt Algorithm and QR decomposition ii) Write python code (without numpy) to decide if Gram-Schmidt Algorithm can be applied to columns of a given matrix A through calculation of rank. The code should print appropriate messages indicating whether Gram-Schmidt isapplicable on columns of the matrix or not.Deliverable(s) : The code that performs the test.Q3. The given coordinates are (0,0), (0,2),(2,0),(2,2) for representing a rectangle/square ,you are expected to find x-shearing where shearing parameter towards x-direction is 2 units. Also you are expected to find y-shearing if the shearing parameter towards y-direction is 3 units. Draw the objects before and after shearing. (Note: You are expected to use matrix representation in calculating the required values)MATLAB questions, 3 simple ones Lets say I have a 100 by 100 matrix consisting random numbers I want to: Find the column that contains most number divisible by 8. There are ‘special sequences’ in the matrix where the first 2 numbers (across the row) add to give the right most number (for example numbers 2, 6, 8…where 2+6 = 8) Find out how many of these ‘special sequences’ there are in the matrix. For eg, 3 numbers are horizontally adjacent in the matrix; 2, 4, 34. The product of these is 272. Find the largest product of 3 adjacent horizontal numbers in the matrix. (loop through matrix) Will give thumbs up!
- Compute the LU factorization of the symmetric part of A(100*100)and verify that your factorization is correct; explain. The symmetric part of a matrix sym(A) is defined as sym(A) =( A+ A^T)/2 Make your best visual demonstration (art contest) that it (the sym(A) matrix) is indeed symmetric. The LU factorization program code is : def lu_factorization(A): 2 """ 3 Performs LU factorization of a matrix A without pivoting. 4 5 Args: 6 A: A square matrix. 7 8 Returns: 9 A tuple (L, U) where L is the lower triangular matrix and U is the upper triangular matrix. 10 """ 11 n = len(A) 12 L = [[0.0] * n for _ in range(n)] 13 U = [[0.0] * n for _ in range(n)] 14 15 for k in range(n - 1): 16 for i in range(k + 1, n): 17 18 mik = A[i][k] / A[k][k] 19 L[i][k] = mik 20 21 for j in range(k + 1, n): 22 A[i][j] = A[i][j] - mik * A[k][j] 23 24 U[k][i] = mik 25 26 for i in range(n): 27 L[i][i] = 1.0 28 29 return L, U 30 31A = [[1.0, 2.0, 3.0], 32 [4.0,…Use the Thomas algorithm to solve the following matrix equation for x1, x2, and x3.17. Let A and B be two n × n matrices. Show that a) (A + B)^t = A^t + B^t . b) (AB)^t = B^t A^t . If A and B are n × n matrices with AB = BA = In, then B is called the inverse of A (this terminology is appropriate because such a matrix B is unique) and A is said to be invertible. The notation B = A^(−1) denotes that B is the inverse of A.