Question

Let A m×n be a given matrix with m > n. If the time taken to
compute the determinant of a square matrix of size j is j to

the power 3, find upper bound on the

a) total time taken to find the rank of A using determinants

b) number of additions and multiplications required to determine
the rank using the elimination procedure.

Answer #1

Let A be an m × n matrix, and Q be an n × n invertible
matrix.
(1) Show that R(A) = R(AQ), and use this result to show that
rank(AQ) = rank(A);
(2) Show that rank(AQ) = rank(A).

Prove the following: Given k x m matrix A, m x n matrix B. Then
rank(A)=m --> rank(AB)=rank(B)

Let A,B be a m by n matrix, Prove that
|rank(A)-rank(B)|<=rank(A-B)

For
n>=3 given the n x n matrix A with elements: A_ij=(i+j-2)^2.
Determine the rank of A.

a)Assume that you are given a matrix A = [aij ] ∈ R n×n with (1
≤ i, j ≤ n) and having the following interesting property:
ai1 + ai2 + ..... + ain = 0 for each i = 1, 2, ...., n
Based on this information, prove that rank(A) < n.
b) Let A ∈ R m×n be a matrix of rank r. Suppose there are right
hand sides b for which Ax = b has no solution,...

Suppose we are given a system Ax = b, with A an n × m matrix.
What can you say about the solution set of the system in the
following cases? Provide a brief explanation.
(i) rank(A) < n
(ii) rank(A) = n
(iii) rank(A) < m
(iv) rank(A) = m

Let M be an n x n matrix with each entry equal to either 0 or 1.
Let mij denote the entry in row i and column j. A
diagonal entry is one of the form mii for some i.
Swapping rows i and j of the matrix M denotes the following
action: we swap the values mik and mjk for k
= 1,2, ... , n. Swapping two columns is defined analogously.
We say that M is rearrangeable if...

In the system AX=b, where A is m x n matrix and rank of A is m,
you are given n vectors and among them p vectors are linearly
dependent (p > m).
Please write down the procedure to reduce the number of
dependent vector by 1.

Finish the following M-file. Run your function on the same
matrix A as given in the lab.
% LU4 - The function in this M-file computes an LU
factorization
% of a 4 x 4 matrix under the assumption that elimination can
be
% performed without row exchanges. % Input: 4 x 4 matrix A;
% Output: lower triangular matrix L and upper triangular matrix
U.
function [L,U] = LU4(A)
L = eye(4);
U = A;
for j= ...
for...

Given a collection of n nuts, and a collection of n bolts, each
arranged in an increasing order of size, give an O(n) time
algorithm to check if there is a nut and a bolt that have the same
size. You can assume that the sizes of the nuts and bolts are
stored in the arraysNUTS[1..n] and BOLTS[1..n], respectively, where
NUTS[1] < ··· < NUTS[n] and BOLTS[1] < ··· < BOLTS[n].
Note that you only need to report whether or...

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