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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.

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
Check out our web site!  http://mathforum.org/dr.math/

Associated Topics:
College Modern Algebra
High School Basic Algebra

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