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Matrix Algebra

Date: 08/28/97 at 21:34:35
From: Corinne Offerman
Subject: Matrices, algebra

      2   4   3             1    2
B = (           )    C = ( -1   -3 )
      1  -1  -2             5   -6

Find BC from this matrix.

I am not sure which formula of matrices to use in this situation. 
Can you please help me?  Thank you.

Date: 08/29/97 at 12:38:42
From: Doctor Anthony
Subject: Re: Matrices, algebra

You should look at a textbook covering the rules of matrix algebra.  
However, the rule for multiplying requires that you combine elements 
of rows from the lefthand matrix with elements of columns of the 
righthand matrix:

     |2    4    3| | 1    2|     |13    -26|
     |1   -1   -2| |-1   -3|  =  |-8     17|
                   | 5   -6|

       (2 x 3)       (3 x 2)  =     (2 x 2)

        |   |        |    |
        |   |_ inner_|    |
        |                 |
        |____ outer ______|

The 'shape' of the lefthand matrix is (2 x 3), that is, it has 2 rows 
and 3 columns. Always quote the number of rows first, then the number 
of columns when specifying the 'shape'. The righthand matrix has shape 
(3 x 2).

Now the rule of multiplication is that number of columns of the 
lefthand matrix must equal the number of rows of the righthand matrix.  
In this example they are both equal to 3, so the product exists. 
If we write down the shapes as I have done below the matrices, we 
see that that the 3's are the 'inner' numbers in the two brackets, 
and these must be equal. The 'shape' of the resulting matrix after 
multiplication is given by the two 'outer' numbers, in this case 
(2 x 2). So the result will have 2 rows and 2 columns.

Finally, to fill in the numbers in the product matrix, we first look 
at the first row, first column position. We get its value by combining 
the elements of the first row of the lefthand matrix with the elements 
of the first column of the righthand matrix. Starting at the first 
element of each we get 2 x 1. 

Now move to second element of each. We get 4 x -1. Then go to third 
element of each. We get 3 x 5. So we must combine 2 - 4 + 15 to get 
13. To get the the value of first row second column of product matrix 
we combine elements of the first row of the lefthand matrix with 
elements of the second column of the righthand matrix. This gives 
2 x 2 + 4 x -3 + 3 x -6 = 4 - 12 - 18 = -26.

The second row of the product matrix is filled in the same manner, but 
now you combine elements of the second row of the lefthand matrix with 
elements of the first and second columns of the righthand matrix.

To fill in the value of any position in the resulting matrix, first 
decide in which row and which column it lies. Then select that row 
from the lefthand matrix, and that column from the righthand matrix, 
and combine them element-by-element in the manner described above.

Having written down the 'shapes', if the two 'inner' numbers are 
not the same, then the product does not exist. Don't waste time 
trying to produce it. Note also that if you swap the matrices round 
from AB to BA you will get a different result, or no result, as the 
product may exist in the AB form but not in the BA form.

-Doctor Anthony,  The Math Forum
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Associated Topics:
College Modern Algebra
High School Basic Algebra

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