Answer to Question #94088 in Linear Algebra for Vanshika

Question

Show that no skew symmetric matrix can be of rank 1

Answer 1

If A is rank 1, then A=uv^T for non-zero column vectors u, v with n entries.

If A is skew-symmetric,

we must have: A^T=−A

Thus:               (uv^T)^T=−uv^T                

Hence:              vu^T=−uv^T

The column space of these matrices is the same. The column space of vu^T is the span of v, whereas the column space of uv^T is the span of u. So, we must have v=ku for some k∈R So, the last equation becomes

kuu^T=−kuu^T

since u≠0, we conclude that k=0, which means that v=0, which means that A=0. But this contradicts our assumption that A has rank 1