Linear Algebra for Computer Science

Question

Let M and N be 2x2 matrices and each is rank 1. What can you say about the renal of M+N?

Answer 1

Generally speaking, for matrices with the same dimension,

rank(M + N) <= rank(M) + rank(N).

So in this case, rank(M*N)<=2.

Ex for rank(M+N) = 2

If

M = [[1, 0], [1,0]]

M = [[0, 1], [0,0]]

Then

M + N = [1, 1], [1, 0]]

and

rank(M) = rank(N) = 1

rank(M + N) = 2

Ex for rank(M+N) = 1

If

M = [[1, 0], [1,0]]

N = M

Then

M + N = [2, 0], [2, 0]]

and

rank(M) = rank(N) = 1

rank(M + N) = 1

Ex for rank(M+N) = 0

If

M = [[1, 0], [1,0]]

N = [[-1, 0], [-1,0]]

Then

M + N = [0, 0], [0, 0]]

and

rank(M) = rank(N) = 1

rank(M + N) = 0