A A and B are any matrices with the same number of rows what

A) A and B are any matrices with the same number of rows.

what can you say about the comparison of

rank of A rank of the block matrix [A B]

B) suppose B= A² How do those ranks compare? Explain your reasoning

C) If A is m by n of rank r, what are the dimensions of these nullspaces?

Nullspace of A

Nulspace of [A A]

Solution

All you can say that rank A <or equal to [A B].( A can have any number r of pivot columns, and these will be a pivot column for [A B] ; but there could be more pivot column among the column of B)

b) now rank A = rank [A A²] (every column of A² is a linear combination of column of A . for instance, if we call Aa₁is the first column of A² part of A² so there are no new pivot columns in the A² part of [AA²])

c) the null space N)A) has dimension n-r, as always.since [A A ] only has r pivot column- the n columns we added are all duplicates- [A A] is an m-by-2n matrix of rank r, and its nullspace N([A A]) has dimension 2n-r.