1 2 Let A =|-5 -8 -1 2 -2 3 1. Give an LU-factorization of A. 2. What is the rank of A? 3. Find basis for the row space of A.
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- Using the below insights: obtain a matrix P such that if A is any matrix with 3 columns, AP is a cyclic shift of the columns of A (namely the first column of A is the second column of AP, second column of A is the third column of AP, and the third column of A becomes the first column of AP). # Let A = a-1, a-2, ..., a-n# x = x-1, x-2, ..., x-n# Ax = x-1*a-1 + x-2*a-2 + ... + x-n*a-n# [x1] [x1]# A = [x2] = [a1 a2 ... an]* [x2] = a1*x1 + a2*x2 + ... + an*xn# [...] [...]# [xn] [xn]If there is a non-singular matrix P such as P-1AP=D, matrix A is called a diagonalizable matrix. A, n x n square matrix is diagonalizable if and only if matrix A has n linearly independent eigenvectors. In this case, the diagonal elements of the diagonal matrix D are the eigenvalues of the matrix A. A=({{1, -1, -1}, {1, 3, 1}, {-3, 1, -1}}) : 1 -1 -1 1 3 1 -3 1 -1 a)Write a program that calculates the eigenvalues and eigenvectors of matrix A using NumPy. b)Write the program that determines whether the D matrix is diagonal by calculating the D matrix, using NumPy. #UsePythonType in Latex **Problem**. Let $$A = \begin{bmatrix} .5 & .2 & .3 \\ .3 & .8 & .3 \\ .2 & 0 & .4 \end{bmatrix}.$$ This matrix is an example of a **stochastic matrix**: its column sums are all equal to 1. The vectors $$\mathbf{v}_1 = \begin{bmatrix} .3 \\ .6 \\ .1 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix} -1 \\ 0 \\ 1\end{bmatrix}$$ are all eigenvectors of $A$. * Compute $\left[\begin{array}{rrr} 1 & 1 & 1 \end{array}\right]\cdot\mathbf{x}_0$ and deduce that $c_1 = 1$.* Finally, let $\mathbf{x}_k = A^k \mathbf{x}_0$. Show that $\mathbf{x}_k \longrightarrow \mathbf{v}_1$ as $k$ goes to infinity. (The vector $\mathbf{v}_1$ is called a **steady-state vector** for $A.$) **Solution**. To prove that $c_1 = 1$, we first left-multiply both sides of the above equation by $[1 \, 1\, 1]$ and then simplify both sides:$$\begin{aligned}[1 \, 1\, 1]\mathbf{x}_0 &= [1 \, 1\, 1](c_1\mathbf{v}_1 +…
- If x=[1 4; 8 3], find :a) the inverse matrix of x .b) the diagonal of x.c) the sum of each column and the sum of whole matrix x.d) the transpose of x.Given a matrix of dimension m*n where each cell in the matrix can have values 0, 1 or 2 which has the following meaning: 0: Empty cell 1: Cells have fresh oranges 2: Cells have rotten oranges So we have to determine what is the minimum time required so that all the oranges become rotten. A rotten orange at index [i,j] can rot other fresh orange at indexes [i-1,j], [i+1,j], [i,j-1], [i,j+1] (up, down, left and right). If it is impossible to rot every orange then simply return -1. Examples: Input: arr[][C] = { {2, 1, 0, 2, 1}, {1, 0, 1, 2, 1}, {1, 0, 0, 2, 1}}; Output: All oranges cannot be rotten. Below is algorithm. 1) Create an empty Q. 2) Find all rotten oranges and enqueue them to Q. Also enqueue a delimiter to indicate beginning of next time frame. 3) While Q is not empty do following 3.a) While delimiter in Q is not reached (i) Dequeue an orange from queue, rot all adjacent oranges. While rotting the adjacents, make sure that time frame is incremented only once. And time frame is…Given a 2-D square matrix: ant mat{3}[{3]={{1, 2 3} {4,586}, {7.8.9FF Write a function transpose which Create a 3*3 matrix trans and store the transpose of given matrix in it.And prxnt the ' transpose, IN C++.
- If a matrix A has size 5x6 and a matrix B has a size 6x4, then what will be the size of a matrix A*B?Consider any 6x6 matrix (only alphabets) and develop as well as demonstrate an algorithm to meet the following requirement- - Create and display all possible strings by combining alphabets present in the chosen matrix based on the following restriction'- * The length of the string should be equal to 6If matrix A is a 2 x 3 matrix, it can be multiplie by matrix B to obtain AB only if matrix B has:A. 2 rowsB. 2 columnsC. 3 rowsD. 3 columns
- Let A be a 3x3 matrix where A2=I3, then det(A) is 1 or -1. Select one: 1-True 2-FalseLet y be a column vector: y = [1, 2, 3, 4, 5, 6] so that y.shape = (6,1). Reshape the vector into a matrix z using the numpy.array.reshape and (numpy.array.transpose if necessary) to form a new matrix z whose first column is [1, 2, 3], and whose second column is [4, 5, 6]. Print out the resulting array z.While applying the Hungarian method on the given matrix in the Assignment problem, we have the following scenario: Suppose, we get the modified matrix, if the number of lines or total number of occupied zero’s is less than number of rows and columns of the matrix, then which one is the right step to be taken to move forward for solving the assignment problem?