Why Does this Example of Matrix Multiplication Work?

Question

I'm fairly new to matrix multiplication and came across an example which looks wrong and I'm wondering if anyone can clarify. Which is said to become the following equation:

But if with matrices you multiply row by column, and then sum, why are the two examples equivalent?

Shouldn't the equation evaluate to

$$\begin{bmatrix} W_{1,1}*x_1 + W_{1,2}*x_1 + W_{1,3} +x_1 \\ W_{2,1}*x_2 + W_{2,2}*x_2 + W_{2,3} +x_2 \\ W_{3,1}*x_3 + W_{3,2}*x_3 + W_{3,3} +x_3 \\ \end{bmatrix}$$

and not

$$\begin{bmatrix} W_{1,1}*x_1 + W_{1,2}*x_2 + W_{1,3} +x_3 \\ W_{2,1}*x_1 + W_{2,2}*x_2 + W_{2,3} +x_3 \\ W_{3,1}*x_1 + W_{3,2}*x_2 + W_{3,3} +x_3 \\ \end{bmatrix}$$

as it does

What would the correct equation be?

Answer 1

The two equations in the picture you posted are not equivalent. The top one is probably a typo.

Answer 2

It looks OK to me.

To do the matrix product, I would go across the Ws and down the Xs. That gets the WX sums.

Then the Bs are added.