Let $S$ be the subspace of any 3 by 3 skew symmetric matrix. Let $T$ be the subspace of all 3 by 3 matrices that are symmetric. Determine the dimension of the intersection of $T$ and $S$ and the dimension of the union of $T$ and $S$ .
What i tried
A symmetric matrix means that $A^T=A$ while skew matrix means $A^T=-A$
For $T$ intersect $S$ the intersection is $0$ since it is not possible for a symmetric and a skew symmetric matrix to intersect.
It is the union part which im unsure I know the ansewer $9$ dimension form my answer key but i couldnt figure out how did they get $9$ I only could think that they multiplied 3 by 3 to get 9. Could anyone explain thanks