What are the multiplication properties of symmetric, anti-symmetric, triangular and diagonal matrices

Question

I am study linar algbera and I am looking for all the multiplication properties of symmetric, anti-symmetric, triangular and diagonal matrices

For example: If $A$ and $B$ are diagonal matrix so $A \times B$ = diagonal matrix as well.

Answer 1

Matrix multiplication is defined for any two matrices with matching dimensions, i.e. for any $A \in \mathbf{K}^{n_a \times m_a}$ and $B \in \mathbf{K}^{n_b \times m_b}$ for any field $\mathbf{K}$, we have the following rule, iff $m_a = n_b$ $$ (A \cdot B)_{ij} = \sum_{k=1}^{m_a} A_{ik} \cdot B_{kj}, $$ for any $i \in \{1, \dots, n_a\}$ and $j \in \{1, \dots, m_b\}$ and thus the resulting matrix $C := A \cdot B \in \mathbf{K}^{n_a \times m_b}$.

E.g. for diagonal matrices the above rule boils down to the multiplication of corresponding elements of the matrix, given the sizes of matrices are the same.