be an identity matrix, and
be a zero matrix. If all five of these matrices have equal dimensions, then we will have the following matrix to matrix multiplication properties:
There are also some matrix to matrix multiplication properties with zero matrices and identity matrices.
Matrix to matrix multiplication property for the zero matrix
Matrix to matrix multiplication property for the identity matrix
Here are some important things to know.
Commutative property fails:
Notice that the commutative property fails when you use matrix to matrix multiplication. For example,
When multiplying a matrix with another matrix, it is not always defined. The product of the two matrices is only defined if the number of columns in the first matrix is equal to the number of rows of the second matrix.