1. Show that the set M of symetric matrices is closed under addition and scalar multiplication.
2. Show that the set W diagonal matrices is not closed under addition and scalar multiplication.
3. Show that the set W of matrices such that transpose(A) = -A is closed under scalar addition and multiplication.
The Attempt at a Solution
ok, for the first one, they have given me the solution but I don't understand why they did that:
Let A and B be in M. then transpose(A)=A and transpose(B)=B.
transpose(A+B)=transpose(A)+transpose(B)=A+B. therefore, M is closed under addition. why do they use matrix transposition?? I don't understand the proof
as for 2 and 3, i have no idea what to do. I don't know how to do it "mathematically", if you can understand what I mean