[Solved] 18.06 Exercise 5-Transposes, permutations, spaces

Problem 5.1: (2.7 #13. Introduction to Linear Algebra: Strang)

1. Find a 3 by 3 permutation matrix with P³ = I (but not P = I).
2. 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:

1. A is symmetric?
2. A is skew-symmetric? (A^T = −A)

Problem 5.3: (3.1 #18.) True or false (check addition or give a counterex­ ample):

1. The symmetric matrices in M (with A^T = A) form a
2. The skew-symmetric matrices in M (with A^T = −A) form a
3. The unsymmetric matrices in M (with A^T = A) form a