# A question about the cross product as related to matrix multiplication

1. Feb 28, 2014

### RufusDawes

I understand that the cross product, in lay mans terms doesn't exist unless we're in 3 dimensions.

When you multiply two matrices together I have been told you get something similar. I hear that this is because a matrix can be treated as a vector.

So if we are talking about measurable things, in the real world does matrix multiplication have any relevance ?

I might be off the mark but what if we were to build a model, based on height weight and income would that be definable in 3 dimensions and therefore if you had two sets of observation and you multiplied them the answer would be nonsense but somehow it has a definition in 3 dimensions as a new matrix ?

if for example, i had a range of multiplies I wanted to apply to each observation or set of observations, stored in a matrix then this would be a scalar multiplication of a matrix onto another matrix ? Does such a thing exist.

Apologies I am not a mathematician so my terminology is buggered but I'm still curious.

Thanks,

2. Feb 28, 2014

### SteamKing

Staff Emeritus
Matrix multiplication is used all the time to design structures, calculate stresses, etc. It's a tool which makes doing certain mathematical calculations easier to organize and handle, especially for computers. It's like asking if regular multiplication has any relevance.

3. Feb 28, 2014

### RufusDawes

okay, so what if I were in grade 1 and didn't understand multiplication ?

Do you have some examples ? Because when I looked at some examples for vector cross product it started to make some sense. Especially when they said it only exists in 3 dimensions.

4. Mar 1, 2014

### SteamKing

Staff Emeritus
The cross product as defined for cartesian coordinates exists and has meaning for 3 dimensions and 7 dimensions.

http://en.wikipedia.org/wiki/Cross_product

The reasons for this are quite mathematically abstract.

The simplest application for matrix multiplication would probably be solving a system of simultaneous linear equations. I can't give you anything simpler than that, just like you would be hard pressed to explain multiplication to a first grader who is just learning to add. Simple matrix manipulations are usually first discussed in algebra classes in middle or high school.

http://www.mathwarehouse.com/algebra/matrix/multiply-matrix.php

Simultaneous linear equations:

http://en.wikipedia.org/wiki/System_of_linear_equations

5. Mar 1, 2014

### RufusDawes

Yes yes but no one ever actually understands what anything means, which is why you have such high drop out rates in later year mathematics courses.

For a system a linear equations it doesn't make sense, because you're multiplying a row by a column.

6. Mar 1, 2014

### Staff: Mentor

The multiplication of a row of a matrix times a column of another matrix (assuming the rows are the same size as the columns) does make sense. It's exactly the same as the dot product of two vectors.

$$\begin{bmatrix} x_1 & x_2 & ... & x_n\end{bmatrix} \begin{bmatrix}y_1 \\ y_2 \\ ... \\ y_n \end{bmatrix}$$
$$=x_1y_1 + x_2y_2 + ... + x_ny_n$$

7. Mar 1, 2014

### SteamKing

Staff Emeritus
Not everyone drops out. Somebody must understand some math some of the time. Otherwise we would be counting on our fingers and drooling at random intervals.

In a nutshell, that's what matrix multiplication is. The particular entry in the product matrix AB is the result of multiplying the row of A by the column of B, for all combinations of the rows and columns in AB.

8. Mar 1, 2014

### RufusDawes

I forgot about that assumption. Now it makes sense.

9. Mar 1, 2014

### Staff: Mentor

Matrix multiplication involves one dot product for each element in the product matrix. Matrix multiplication doesn't have anything to do with the cross product.

10. Mar 1, 2014

### nmf77

true, unless you form a skew symmetric matrix from a vector, which is a common way to compute cross products