Do problems 21 and 22, page 127, of your textbook.
An nxn matrix A is said to be symmetric if its ij-th entry equals its ji-th entry. In other words a matrix is symmetric if it remains unchanged when it is flipped about its main diagonal. Notice that this condition does not impose any restictions on the diagonal entries.
An nxn matrix A is said to be anti-symmetric if its ij-th entry is equal to the negative of its ji-th entry. So now a matrix is anti-symmetric if it is left unchanged when each entry is multiplied by -1 and then it is flipped about its main diagonal.
Question 1: Rephrase the above two definitions using the transpose of the matrix.
Question 2: If a matrix A is anti-symmetric, what can you say about the diagonal entries?
Question 3: If A is an nxn anti-symmetric matrix and n is odd, then the determinant of A can be determined. What is it? (Try a number of examples using LINALG. Choose "ANTISYMETTRIC" when creating the matrices.) Can you prove your answer?
Question 4: Let A be an anti-symmetric 2x2 matrix. What can you say about its determinant?
Question 5: Experiment with anti-symmetric 4x4 matrices. What would your guess be about determinants of an nxn anti-symmetric matrices when n is even?
