Question

a)Assume that you are given a matrix A = [aij ] ∈ R n×n with (1...

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, which of following expression(s) is/are correct: r < m, r = m, r > m.

Now, consider the linear system AT y = 0. Do you think this linear system can have non-zero solutions, that is y 6= 0 such that AT y = 0. Give justification for all your answers

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