Closedness of Matrices Under Addition & Multiplication

[fishingspree2]

Homework Statement

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

thank you

 

[Mark44]

For 1, what does it mean for a matrix to be symmetric? I.e., what's the definition? That's where the transpose comes in.
For 2, if the matrices in set W are a specified size, addition and scalar multiplication are closed. Since there is no mention of the size of the matrices in W, I interpret this to mean that W includes diagonal matrices of different sizes.

For 3, take two matrices (A and B) in your set W and show that A + B is in W. Also show that if A is in W, then kA is also in W, where k is a scalar. BTW, you want to show that W is closed under addition and scalar multiplication, not scalar addition and multiplication.