Skew-Symmetric Matrices: An n × n matrix J is called skew-symmetric if JT = - J.
(a) Show that every diagonal entry of a skew- symmetric matrix is zero.
(b) Write down an example of a nonsingular skew- symmetric matrix.
(c) Can you find a regular skew-symmetric matrix?
(d) Show that if J is a nonsingular skew-symmetric matrix, then J-l is also skew-symmetric. Verify this fact for the matrix you wrote down in part (b).
(e) Show that if J and K are skew-symmetric, then so are J1, J + K, and J - K. What about J K?
(f) Prove that if J is a skew-symmetric matrix, then vTJ v = 0 for any vector v.