Properties of matrix multiplication - Matrices

Notes:

Let $X, Y, Z$ be matrices, $I_n$ be an identity matrix, and $O_n$ be a zero matrix. If all five of these matrices have equal dimensions, then we will have the following matrix to matrix multiplication properties:
Associative property
$(XY)Z=X(YZ)$

Distributive property
$X(Y+Z)=XY+XZ$
$(Y+Z)X=YX+ZX$

There are also some matrix to matrix multiplication properties with zero matrices and identity matrices.
Matrix to matrix multiplication property for the zero matrix
$OX=O$ or $XO=O$
Matrix to matrix multiplication property for the identity matrix
$XI_n=X$ or $I_n X=X$

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, $XY \neq YX$.

Dimension property: 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.