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Matrices Explained in 5 Minutes

Matrices Explained in 5 Minutes

Take a few numbers and arrange them in a table like this..

3 5 1 6
2 8 6 7
0 4 5 2

Now, to make the table look pretty enclose it in giant brackets ( ). The brackets are not needed for any mathematical reason, they're just a reminder that the entire table should be considered as a single mathematical object.

You just made a matrix! It has 3 rows and 4 columns of numbers, so it's a 3x4 matrix. The numbers in a matrix are called the elements or terms of the matrix. So our 3x4 matrix has 12 elements. Of course, you can make matrices of any size. If the number of rows and columns are the same it's called a square matrix. Here's an example..

2 3 0
2 1 6
1 4 3

This is a square 3x3 matrix.

A matrix with only one row or one column is a vector, like this..

One row..

(2 3 1)

One column..


Of course, writing down new mathematical objects is easy. The harder part is defining some operations you can do with the objects.

Square Matrices can be added, just add corresponding terms to make a new matrix, like this example..

5 1
3 2


2 6
4 7


5+2 1+6
3+4 2+7

which is the matrix..

7 7
7 9

So addition is easy. But the operation which makes matrices very useful mathematical objects is the multiplication operation. This operation is subtle, and it's not just multiplying corresponding terms. To understand matrix multiplication, and also to see why it's so useful, consider what happens when a matrix multiplies a vector. After all, a vector is just a special case of a matrix.

So here's a matrix..

3 5 1
2 8 6
0 4 5

and here's a vector written as a one column matrix..


Now let's multiply..

Take the first row of the matrix (3 5 1), write it vertically, like this..


Now just multiply it term by term with the vector and add all the results, like this..


That's the first element of our result, so our result looks like..

To get the second term of the result repeat the process, but this time use the second row of the matrix (2 8 6) to get..


So now our result is..


To get the final term of the result use the third row of the matrix (0 4 5) to get..


So the final answer is the vector..


We're done! We just multiplied a vector by a matrix and the answer was another vector. So, one important job that matrices do is transform vectors. They can rotate them, they can stretch them, or they can do both at the same time. It turns out this is a very useful operation and is the reason matrices are so useful.

Of course, the table structure of a matrix lends itself to a very elegant notation. Suppose we have an nxm matrix, that is one with n rows and m columns. Then we can denote the general term of the matrix by aij, which is the number in the i'th row and j'th column. So the entire matrix could be written simply as..

{aij} where i=1,2,..,n and j=1,2,..,m

This is far easier than writing out a giant nxm table of numbers! Plus, this notation turns out to be very powerful, for example, it let's us write the product of two matrices in a very elegant way. Suppose {aij} and {bij} are two nxm matrices, then the ij'th term of the product is..

sum(aik*bkj) for k=1,2,...,m

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Content written and posted by Ken Abbott
Internet Marketing Consultant