I'm having a bit of trouble with the following question.
Suppose $A$ is a square matrix.
a) Show that the matrix $B = A+A^T$ is symmetric.
Not sure how to do this. But here is my attempt. Well, let $A$ have a size of $n\times n$. That means $A^T$ must also have a size of $n$ by $n$ by the transpose property number of columns and rows are swapped. Therefore $B$ must also have a size $n$ by $n$. That's all I have at the moment.
b) Show that $C=AA^T$ is symmetric
I also have approached this like a) But not sure.
c) A matrix $M$ has a property that $M^T = - M$ (skew symmetry). Show that $D = (A - A^T)$ is a skew symmetric matrix.
d) Can you show how to write any square matrix as the sum of a symmetry and skew symmetric matrix.