Assume A is k x n-matrix and B is n x l-matrix and AB = 0. Prove that rank(A) + rank(B) < n.
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Q: Assume A is k × n-matrix and B is n × l-matrix. Prove that rank(AB) < rank(A) and rank(AB) < rank(B)
A: Assume A is k x n-matrix and B is n x l- matrix. Prove that rank(AB)
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Q: ) Assume A is k × n-matrix and Q is n x n-invertible matrix. Prove that rank(AQ) = rank(A).
A: Solution:
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- 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.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 +…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 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?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.Write a matlab code for Creating 5x5 matrix with rank 4 and generate 2 vectors each that are not in a. Column space b. Rowspace c. Left null space d. Right null space Explain how will you verify your answer for various cases in above problem
- What is the worst-case running time complexity of matrix substraction? select one: a.O(n^2.5) b.O(2n) c.O(n^2) d.O(3^n)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++.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?
- Write down the tensor expression for the following matrix operations. Where A,B are 3 × 3, C is 3 × 4, and D is 4 × 7 matrices. And for a matrix M, its i-th row j-th column component will be denoted as M_ij . (a) det(A · B^T ).(b) Tr[A · B].(c) B · C · D, and explicitly spell out all the index summation.(d)δ_ii =?(e) ϵ_ijkϵ_ijk =?Is a rank of a matrix can be zero and what is nullity of a matrix?Matlab question3. Explain why the next matrix in Floyd’s algorithm can be written over its predecessor (i.e., when calculating D(k) from D(k-1), we do not need another matrix but can change D(k-1) into D(k).