1. ## The coefficients-vectors method

Ok, I understand how to get it this far. But how are they getting this?

2. Hello there!

I'm not sure what you are trying to do, multiplication or dot product (the single dot usually indicates dot product).

But the matrix below is the result of multiplying those two matricies. Is that what you are having a problem with?

3. nm, i just needed to sit back and look at it more.

4. It makes no difference which method you use to multiply matrices, whichever one you understand the best. The important thing is that you know how to multiply them. The one I use is the standard dot product between the rows and coloums.

5. Hello, Grunt!

Someone taught you a very messy way of multiplying matrices.
. . Shame on them!

ThePerfectHacker referred to a "standard dot product".
. . I thought everyone used it.

Let's multiply "a row times a column".
Code:
            | 4 |
[ 1 2 3 ] · | 5 |  =  (1)(4) + (2)(5) + (3)(6)
| 6 |

=  4 + 10 + 18  =  32
I hope you see what pairs are being multipled.

Code:
                   | 3  1 |
|  1  0  2 |   |      |
|          | · | 2  1 |
| -1  3  1 |   |      |
| 1  0 |
We will multiply each row in the first matrix
. . by each column in the second matrix.
Here we go . . .

Row-1 times Column-1
Code:
                | 3 |
| 1 0 2 | · | 2 |  =  (1)(3) + (0)(2) + (2)(1)  =  5
| 1 |
In Row-1, Column-1 of the product matrix (upper left).

Row-1 times Column-2
Code:
                | 1 |
| 1 0 2 | · | 1 |  =  (1)(1) + (0)(1) + (2)(0)  =  1
| 0 |
This goes in Row-1, Column-2 of the product (upper right).

Row-2 times Column-1
Code:
                | 3 |
|-1 3 1 | · | 2 |  =  (-1)(3) + (3)(2) + (1)(1)  =  4
| 1 |
This goes in Row-2, Column-1 of the product (lower left).

Row-2 times Column-2
Code:
                | 1 |
|-1 3 1 | · | 1 |  =  (-1)(1) + (3)(1) + (1)(0)  =  2
| 0 |
This goes in Row-2, Column-2 of the product (lower right).

      | 5  1 |
| 4  2 |