Let a 3 × 3 matrix A be such that for any vector of a column v ∈ R3 the vectors Av and v are orthogonal. Prove that At + A = 0, where At is the transposed matrix.

!

Moderator Note

We do not do other people's homework for them. Please show some attempt at an answer and explain here you are stuck. 

 

22 hours ago, hypervalent_iodine said:

!

Moderator Note

We do not do other people's homework for them. Please show some attempt at an answer and explain here you are stuck. 

 

Its not a homework, lol. Just an interesting problem, that i don't quite understand. I think that the only option is that A matrix is zero matrix or maybe i have forgotten some theorem  or criteria

Well, as stated the first part has only one solution, the zero matrix; on the other hand the zero matrix is skew-symmetric..

The first part could be modified but why bother, its gonna be a rotation of \( \pi/2 \) radians in some direction, i.e. the solution space could be visualized as a circle with every point a matrix.