How Do You Correctly Set Up and Multiply Matrices for Business Sales Analysis?

[noobshoob]

I don't know if this is the correct place to post this thread, but I have come across a few problems which I do not know how to approach.

1. A shop sells TVs at $300, DVDs at $20 and Radios at $50. Put these into a row matrix.
Here is what I did:

300
20
50

Is this correct? Should it be [300, 20, 50] (horizontally, if it matters)?

2. On Monday they sell 5 TVs, 2 DVDs and 3 Radios.
On Tuesday they sell 7 TVs, 3 DVDs, and 2 Radios.
On Wednesday they sell 5 TVs, 6 DVDs and 4 Radios.
Set out the costs and the quantities into 2 matrices.
Here is what I did:

300      5  7  5
20       2  3  6
50       3  2  4

3. Calculate the product of the cost and quantity matrices to form the income matrix.
Should I multiply them? Use dot product?

 

[AssyriaQ]

For part 1, no it is not correct. What you have written is a column matrix. Your other suggestion is correct.

Have you done matrix multiplication before? You cannot multiply a 3x1 matrix and a 3x3 matrix. However, you can multiply a 1x3 and a 3x3. That is why it matters whether your write a row or a column matrix.

 

[noobshoob]

I've multiplied 2 2x2 matrices before. So it is okay for me to multiply [300, 20, 50] with the 3x3 matrix?

How about questions 2 and 3? Is my 3x3 matrix correct or should the numbers be in a different order?

 

[HallsofIvy]

Well, if you are multiplying with "price matrix" on the left, then it must be a row matrix:
$$\begin{bmatrix} 300 & 20 & 50\end{bmatrix}\begin{bmatrix}5 & 7 & 5 \\ 2 & 3 & 6 \\ 3 & 2 & 4\end{bmatrix}$$

If you choose, instead, to multiply with price on the right, then it would be a column matrix, but also with the transpose of the price matrix:
$$\begin{bmatrix}5 & 2 & 3 \\ 7 & 3 & 2 \\ 5 & 6 & 4\end{bmatrix}\begin{bmatrix} 300 \\ 20 \\ 50\end{bmatrix}$$

Edit:
It occurs to me after writing this that how you set the matrix multiplication up depends on what, exactly, "income matrix" means. Is it income summed over different kinds of product per day or summed over the three different days per kind of product.

The total would be either of those "dot product" [1 1 1].

 

[noobshoob]

Your first suggestion is what I'm going to go with I think as it seems to fit the question.
However, is my 3x3 matrix okay? Is the order of the values like this...

5  7  5
2  3  6
3  2  4

...okay, or should it be:

5  2  3
7  3  2
5  6  4

It doesn't specify whether it is the total income or just the income per day, but since the question prior to the last asks me to create a matrix to hold all the income of the 3 days then I guess it would be logical to calculate the total. You say that I must use the dot product on the two matrices? Why would this be?