Exercises on transposes, permutations, spaces
Problem 5.1: (2.7 #13. Introduction to Linear Algebra: Strang)
a) Find a 3 by 3 permutation matrix with P3 = I (but not P = I).
b) Find a 4 by 4 permutation P with .
Problem 5.2: Suppose A is a four by four matrix. How many entries of A can be chosen independently if:
a) A is symmetric?
b) A is skew-symmetric? (AT = −A)
Problem 5.3: (3.1 #18.) True or false (check addition or give a counterex ample):
a) The symmetric matrices in M (with AT = A) form a subspace.
b) The skew-symmetric matrices in M (with AT = −A) form a subspace.
c) The unsymmetric matrices in M (with AT = A) form a subspace.
