How to evaluate a $A_{x\times y}B_{y\times z}$ where $ A$ and $B$ are matrices, $x\neq z$ I know how to evaluate a Cx,y * Dy,x (C rows are equals to D columns), but how do I evaluate a matrix multiplication in which the involved matrices (A and B) have respectively different number of rows and columns?
Here is the "mask": $A_{x\times y}  B_{y\times z}$ (x,y in A and y,z in B are the matrices order).
EDIT:
I wanna know how to evaluate a multiplication of matrices that have A number of columns = B number of rows, but A number of rows != B number of columns.
You usually multiply the elements of the A rows with the elements of B columns, but if the number of rows are different from the number of columns some elements would remain, what should I do?
 A: Your getting what the indices denote mixed up: $C_{x\times y}\times D_{y \times x} = E_{x\times x}$ denotes the product of a matrix $C$ with $x$ rows and $y$ columns, times a matrix $D$ having $y$ rows and $x$ columns. This results in a matrix $E_{x\times x}$ which has $x$ rows and $x$ columns: i.e. $E$ is then a square matrix.
Multiplication of matrices $A\cdot B$ is defined if and only if the number of columns of $A$ is equal to the number of rows of $B$.
So in your case $A_{\large {\bf x}\times\color{blue}{\bf y}} \cdot B_{\large \color{blue}{\bf y}\times {\bf z}}$ is defined, and results in a matrix $C_{\large \bf x\times z}$.
Here the first index denotes the number of rows of a matrix, and the second index denotes the number of columns of a matrix. We see that multiplication is defined because there are $\color{blue}{\bf y}$ columns in $A$, and $\color{blue}{\bf y}$ rows in $B$.
So, e.g., $A_{x\times y}$ denotes a matrix with $x$-rows, and $y$-columns, and $x\times y$ is called the dimension of the matrix $A$.
Example:
Lets multiply: $A_{2\times 3}\cdot B_{3\times 3} = \begin{bmatrix}1 & 2 & 3 \\ 1 & 2 & 0\end{bmatrix} \cdot \begin{bmatrix} 0 & 1 & 0\\ 2 & 0 & 1\\ 1 & 0 & 2\end{bmatrix}$
We will obtain a product $$ \begin{align}A\cdot B =C_{2\times 3} & = \begin{bmatrix} (1\cdot 0 + 2\cdot2 + 3\cdot 1) & (1\cdot 1 + 2\cdot 0 + 3\cdot 0) & (1\cdot 1 + 2\cdot 0 + 3 \cdot 2)\\ (1\cdot 0 + 2\cdot 2 + 0\cdot 1) & (1\cdot 1 + 2\cdot 0 + 0\cdot 0) & (1\cdot 0 + 2\cdot 1 + 0\cdot 2)\end{bmatrix} \\ \\ &= \begin{bmatrix} 7 & 1 & 7 \\ 4 & 1 & 2 \end{bmatrix}\end{align}$$
