Are These Sets of Matrices Closed Under Multiplication?

[summer]

A set S of (necessarily square) matrices is said to be closed under multiplication if AB∈ S whenever A, B ∈ S.

Which of these matrices are closed under multiplication?

Circulant matrices
Upper triangular matrices
Hessenberg matrices

My trouble: How do I go about figuring this one out?

 

[RicoB]

"A set S of (necessarily square) matrices is said to be closed under multiplication if AB∈ S whenever A, B ∈ S."

Perhaps you mean that the multiplication of 2 matrices of the same type will give as result a matrix of the same type.If that's so, then all triangular matrices are closed under multiplication (it can be proved, just have to use the definitions of triangular matrices and matrix pultiplication).
I'm not familiar with the other types os matrices.

Best regards

 

[HallsofIvy]

you mean "which of these sets of matrices is closed under multiplication".

If you multiply two circulant matrices is the result always a circulant matrix?

If you multiply two Hessenberg matrices is the result always a Hessenberg matrix?