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Multiplying Matrices and Identifying Dimensionality

```Date: 07/16/2003 at 13:03:39
From: Rick
Subject: Multiplying Matrices and Identifying Dimensionality

I understand adding and subtracting matrices, but not multiplying
them. I have this question, how do I do this?

[-2  4  -6]    [1  -1  1]
[ 1  -1  0]    [3  0  -2]
[ 0   2  5]    [4  -6  7]

Also, how can I find the dimensionality of the following matrix?

[2]
[4]  [3  1  7]
[8]

would 3x1  and 1x3 be correct and therefore they cannot be multiplied
because the inner dimensionalities are different?

Thank you very much,
Rick
```

```
Date: 07/16/2003 at 23:11:39
From: Doctor Peterson
Subject: Re: Multiplying Matrices and Identifying Dimensionality

Hi, Rick.

Your second example is a 3x1 times a 1x3, so the inner dimensions ARE
the same (both 1), and they CAN be multiplied. Just take each row of
the first and multiply it by each column of the second; in this case
each is just a single pair of numbers:

[ 3   1   7 ]

[2]  [2*3 2*1 2*7]   [ 6  2 14]
[4]  [4*3 4*1 4*7] = [12  4 28]
[8]  [8*3 8*1 8*7]   [24  8 56]

Do you see how I did it? In this case, it looks like a multiplication
table, each row and column of the product being the product of the
number in that row of the first matrix and the number in that column
of the second matrix, which I wrote above for clarity.

To multiply matrices with more than one column, you have to multiply
and add, rather than just multiply. For a simple example, look at

[1 2] [5 3]
[3 4] [2 4]

I can write them as above,

[   5       3   ]
[   2       4   ]

[1 2]  [1*5+2*2 1*3+2*4]   [ 9 11]
[3 4]  [3*5+4*2 3*3+4*4] = [23 25]

Here I "multiplied" the row [1 2] by the column [5,2] by multiplying
corresponding pairs and adding the products: 1*5 + 2*2. Repeat that
for all pairs of row and column, and you're done. Note that the inner
dimensions have to agree, or you can't pair up numbers to multiply.

Here is a site that covers matrices nicely, which may help you when

S.O.S. Math - Matrix Algebra
http://www.sosmath.com/matrix/matrix.html

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 07/17/2003 at 09:26:47
From: Rick
Subject: Multiplying Matrices and Identifying Dimensionality

This makes sense now, but one more question. Would I do this one below
the same way as my second one I showed you last time?

[1]
[5]
[2 5 9]    [4]

Would it be....

2*1  2*5   2*4
5*1  5*5   5*4
9*1  9*5   9*4

Would this above be the correct way to do it? And then multiply the
numbers of course.

Thanks.
```

```
Date: 07/17/2003 at 09:36:53
From: Doctor Peterson
Subject: Re: Multiplying Matrices and Identifying Dimensionality

Hi, Rick.

You multiplied in the wrong order. (Matrix multiplication is NOT
commutative.) When you multiply a 1x3 and a 3x1, you get a 1x1; you
multiplied the 3x1 and the 1x3 and got a 3x3.

Here is the correct work:

+-----------[1]
| +---------[5]
| | +-------[4]
| | |
[2 5 9] [2*1+5*5+9*4] = [63]

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Linear Algebra

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