Question

Prove that the rows/columns of a skew-symmetric matrix a of odd order are linearly dependent.

Answer 1

Hello.


Problem 564

Let A and B be n×n skew-symmetric matrices. Namely AT=−A and BT=−B.

(a) Prove that A+B is skew-symmetric.

(b) Prove that cA is skew-symmetric for any scalar c.

(c) Let P be an m×n matrix. Prove that PTAP is skew-symmetric.

(d) Suppose that A is real skew-symmetric. Prove that iA is an Hermitian matrix.

(e) Prove that if AB=−BA, then AB is a skew-symmetric matrix.

(f) Let v be an n-dimensional column vecotor. Prove that vTAv=0.

(g) Suppose that A is a real skew-symmetric matrix and A2v=0 for some vector v∈Rn. Then prove that Av=0.



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