Given matrices $A$ and $B$, the Hadamard product $A\circ B$ is given by $(A\circ B)_{ij}=A_{ij}\cdot B_{ij}$ (if you ask an ordinary middle schooler, this would be the "most natural" definition of matrix multiplication, haha.)
Does anyone know if it is possible to represent $A\circ B$ as a combination of other widely used matrix operations, such as addition, multiplication, taking determinant and taking inverse?