Operations Research : Applications and Algorithms
4th Edition
ISBN: 9780534380588
Author: Wayne L. Winston
Publisher: Brooks Cole
expand_more
expand_more
format_list_bulleted
Expert Solution & Answer
Chapter 2.5, Problem 8P
Explanation of Solution
a.
Inverse of the given matrix:
We know that for any square matrix
Therefore, we have,
Hence, we get
Explanation of Solution
b.
Obtaining the inverse:
Suppose
Then we have,
Now, we have,
Explanation of Solution
c.
Obtaining the inverse:
Suppose
Then, we have,
Now, we have,
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
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 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…
Chapter 2 Solutions
Operations Research : Applications and Algorithms
Ch. 2.1 - Prob. 1PCh. 2.1 - Prob. 2PCh. 2.1 - Prob. 3PCh. 2.1 - Prob. 4PCh. 2.1 - Prob. 5PCh. 2.1 - Prob. 6PCh. 2.1 - Prob. 7PCh. 2.2 - Prob. 1PCh. 2.3 - Prob. 1PCh. 2.3 - Prob. 2P
Ch. 2.3 - Prob. 3PCh. 2.3 - Prob. 4PCh. 2.3 - Prob. 5PCh. 2.3 - Prob. 6PCh. 2.3 - Prob. 7PCh. 2.3 - Prob. 8PCh. 2.3 - Prob. 9PCh. 2.4 - Prob. 1PCh. 2.4 - Prob. 2PCh. 2.4 - Prob. 3PCh. 2.4 - Prob. 4PCh. 2.4 - Prob. 5PCh. 2.4 - Prob. 6PCh. 2.4 - Prob. 7PCh. 2.4 - Prob. 8PCh. 2.4 - Prob. 9PCh. 2.5 - Prob. 1PCh. 2.5 - Prob. 2PCh. 2.5 - Prob. 3PCh. 2.5 - Prob. 4PCh. 2.5 - Prob. 5PCh. 2.5 - Prob. 6PCh. 2.5 - Prob. 7PCh. 2.5 - Prob. 8PCh. 2.5 - Prob. 9PCh. 2.5 - Prob. 10PCh. 2.5 - Prob. 11PCh. 2.6 - Prob. 1PCh. 2.6 - Prob. 2PCh. 2.6 - Prob. 3PCh. 2.6 - Prob. 4PCh. 2 - Prob. 1RPCh. 2 - Prob. 2RPCh. 2 - Prob. 3RPCh. 2 - Prob. 4RPCh. 2 - Prob. 5RPCh. 2 - Prob. 6RPCh. 2 - Prob. 7RPCh. 2 - Prob. 8RPCh. 2 - Prob. 9RPCh. 2 - Prob. 10RPCh. 2 - Prob. 11RPCh. 2 - Prob. 12RPCh. 2 - Prob. 13RPCh. 2 - Prob. 14RPCh. 2 - Prob. 15RPCh. 2 - Prob. 16RPCh. 2 - Prob. 17RPCh. 2 - Prob. 18RPCh. 2 - Prob. 19RPCh. 2 - Prob. 20RPCh. 2 - Prob. 21RPCh. 2 - Prob. 22RP
Knowledge Booster
Similar questions
- Consider the chain of matrices below. M = M1 x M2 x M3 x M4 [15 x 5] [5 x 25] [25 x 30] [30 x 45] (a) Show the complete table used by the dynamic programming algorithm for the matrix chain problem.arrow_forwardLet A be an m × n matrix with m > n. (a) What is the maximum number of nonzero singular values that A can have? (b) If rank(A) = k, how many nonzero singular values does A have?arrow_forwardWhile 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?arrow_forward
- 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?arrow_forwardIf 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. #UsePythonarrow_forwardGiven 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++.arrow_forward
- Type 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 +…arrow_forwardsubject : analysis of algorithm Q.No.1: Consider the following chain of matrices having matrices A, B, C and D. You have to consider the digits of your Registration Number in the order of the matrix as given. Add 2 to the digit if its zero. For example, your Reg_No. 19-Arid-797 has last digit 7, 2nd last digit 9 and 3rd last digit 7. A B C 2 X last digit last digit X 2nd last digit 2nd last digit X 3nd last digit D 3rd Last digit X 4 What will be the minimum number of multiplication to multiply these matrices? Show the order of multiplication as well.arrow_forwardLet f ∈ C+ 2π with a zero of order 2p at z. Let r>p and m = n/r. Then there exists a constant c > 0 independent of n such that for all nsufficiently large, all eigenvalues of the preconditioned matrix C−1 n (Km,2r ∗ f)Tn(f) are larger than c.arrow_forward
- We define a magic square to be an matrix of distinct positive integers from to where the sum of any row, column, or diagonal of length is always equal to the same number: the magic constant. You will be given a matrix of integers in the inclusive range . We can convert any digit to any other digit in the range at cost of . Given , convert it into a magic square at minimal cost. Print this cost on a new line. Note: The resulting magic square must contain distinct integers in the inclusive range . Example $s = [[5, 3, 4], [1, 5, 8], [6, 4, 2]] The matrix looks like this: 534158642 We can convert it to the following magic square: 834159672 This took three replacements at a cost of . Function Description Complete the formingMagicSquare function in the editor below. formingMagicSquare has the following parameter(s): int s[3][3]: a array of integers Returns int: the minimal total cost of converting the input square to a magic square Input Format Each of the lines…arrow_forwardLet 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.arrow_forwardLet Tmn be a BTTB matrix with a generating function f(x, y) ∈C2π×2π. Let λmin(Tmn) and λmax(Tmn) denote the smallest and largest eigenvaluesof Tmn, respectively. Then we havefmin ≤ λmin(Tmn) ≤ λmax(Tmn) ≤ fmax,where fmin and fmax denote the minimum and maximum values of f(x, y), respectively. In particular, if fmin > 0, then Tmn is positive definite.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole
Operations Research : Applications and Algorithms
Computer Science
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Brooks Cole