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COMP 4560
Computer Graphics for Computer Systems Technology
Matrix Multiplication
Matrix multiplication plays a major role in the formalism we will set up in the course to carry out geometric
and viewing transformations. This document is a brief review of how matrix multiplication is done, with a few
examples or exercises for you to practice with if necessary.
A matrix is a rectangular table of numbers. For example:
1
6 2 9


A  5 0 3 8 
7 5 12 4 
is a matrix with three rows and four columns (we would call this a 3 by 4 matrix). Matrices are usually
denoted in print by bolded upper case characters. The individual entries in a matrix are called its elements,
and are often referred to using a subscript notation indicating the row and column in which they are found.
Thus, the '-5' in the above matrix is also called A32, because it is the element in the third row and second
column of the matrix. Matrices with the same number of rows and columns are referred to as square
matrices.
A variety of arithmetic operations are defined for matrices. For this course, multiplication of one matrix by
another is the most useful. The product of two matrices, A, and B, is a new third matrix C:
A

B

C
(1)
nxm
mxk
nxk
For the multiplication to be possible, the number of columns in the first factor must equal the number of rows
in the second factor. When this is true, the product can be formed, and the result is a matrix which has the
same number of rows as the first factor, and the same number of columns as the second factor.
To form element Cjk of the product, we simply multiply together corresponding elements of row j of matrix A
and column k of matrix B, and add up the m products formed that way. That is
C jk  Aj 1B1k  Aj 2B2k  Aj 3B3k 
m
Ajm Bmk   AjL BLk
(2)
L 1
To form one element of the product requires m multiplications, and there nk elements to be formed, so the
entire operation requires nkm multiplications altogether. You could regard the calculation of the element C jk
as the "product" of row j of matrix A by column k of matrix B. Since elements of row j of A are multiplied by
the corresponding elements of column j of matrix B, you can see why the number of columns in matrix A
(which is equal to the number of elements in each row of matrix A) must equal the number of rows in matrix
B (which is equal to the number of elements in each column of matrix B).
Example: As a numerical example, note that:
7 2 
 79 50 

 9 6  

9 3   8 4  105 66 
  57 30 
 1 6  


In detail, this result comes from:
David W. Sabo (2000)
Matrix Multiplication
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for the 11-element:
for the 12-element:
for the 21 element:
for the 22 element:
for the 32 element:
for the 33 element:
7 x 9 + 2 x 8 = 63 + 16 = 79
7 x 6 + 2 x 4 = 42 + 8 = 50
9 x 9 + 3 x 8 = 81 + 24 = 105
9 x 6 + 3 x 4 = 54 + 12 = 66
1 x 9 + 6 x 8 = 9 + 48 = 57
1 x 6 + 6 x 4 = 6 + 24 = 30
(row 1 times column1)
(row 1 times column 2)
(row 2 times column 1)
(row 2 times column 2)
(row 3 times column 1)
(row 3 times column 2)

Remarks
i) matrix multiplication is not commutative  that is, the order in which the matrices appear in the product is
important. For numbers, we know that 2 x 3 is the same as 3 x 2. However, in general, for matrices
AB  BA
First of all, even when AB makes sense and can be calculated, the dimensions of the two matrices may not
allow the formation of the product BA (see exercise 2 below). When BA can be formed, in general, it will
have different elements than does AB, and may even have a different shape (see exercises 1 and 3
below).
ii) matrix multiplication is associative  that is, when you have the product of three or more matrices, it does
not matter in which order you carry out the multiplications, as long as you keep the matrices in the original
order. Thus, to calculate, say, ABC, you can first form AB and then multiply this result from the right by
matrix C, or, you can first form BC and then multiply this result by A from the left. The final result will be the
same (see exercise 4 below).
iii) The square matrix with diagonal elements equal to 1 and all other elements equal to zero is called the
identity matrix for multiplication, denoted by I, the upper case "eye". Multiplication of a matrix A by the
appropriately dimensioned I from either the right or the left gives the result A. Thus, multiplication by I for
matrices is like multiplying by the number 1 in ordinary arithmetic.
iv) If A is a square matrix and you can find a second matrix, B, such that AB = I, then the matrix B is called
the inverse matrix of A, and is usually denoted by a superscript '-1' as in A-1. For square matrices, A, the
inverse matrix A-1, when it exists, works in this way when multiplied from the right or the left. Thus
AA-1 = A-1A = I
You can see that this means that whatever A accomplishes, A-1 reverses, because the net effect of applying
the two matrices in sequence in no change. Since we are using matrix multiplication to carry out geometric
manipulations of shapes, the inverse matrix is something like the "undo" operation in computerese.
Exercises
7 6 
 and B 
6 1 
1. Given A  
10 8 

 , compute AB and BA.
 5 7
 1 9 7 


5
1 9
2. Given A  
and B 
 5 4 1 


 7 6 4 
 3 
 
3. Given A  9
  and B  4 7
 2
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1 5 


5 10  , compute AB and BA.
 1 9 
2 , compute AB and BA.
Matrix Multiplication
David W. Sabo (2000)
 3 4 2 8 
 6 4 5 


4. Given A  
 , B   7 5 6 9 and C 
7 3 
8
 2 1 0 4 
 7 1 


 2 8  , find the product ABC in
 6 5 


9 
 0
two different ways. First multiply B by C to form BC, and then multiply this from the left to get ABC.
Then, multiply A by B to get AB and then multiply this by C from the right to get ABC. Compare your two
final results.
 2 3 9 
 7 75 69 




5. Given A  2 8 1 and B  4
4 16  , compute AB and BA.



 0 9 2
 18 18 10 
 1 3 4 
 7 6 8 




2 5  and B   2 5 7  , compute AB and BA.
 3 5 3 
 8 10 3
6. Given A  10

7. If you still haven't had enough, notice that the matrices in exercise 4 above have dimensions that allow
you to calculate two other triple products: BCA and CAB. Do so.
Answers
 37 22 


 40 14 
118 52 
 19 116  , BA cannot be formed because
B

A

A

B

1. A  B  
,
;
2.



 24 56 
 65 55 
 77 23 


 27 61
you cannot multiply a 3 x 2 matrix by a 4 x 3 matrix.
 12

3. A  B  36

 8
 56 9
A B  
 79 0
21
6 
 17 97 
,


63 18 B  A  55 ; 4. B  C   75 84  , A  B  C 
 16 46 
14 4 
 482 476 

,
 709 326 
12 32
 , you should get the same result for ABC whichever method you use.
26 13 
0
0 
 164


5. A  B  B  A 
164
0   these two matrices do commute.
 0
 0
0
164
61 17 
 33
 91 49 82




6. A  B  114 100 51 , B  A  69 19 38




 35 73 50 
99 11 73
 878 747 376


7. B  C  A  222 288 123 , C  A  B 


 464 386 218 
237 
 471 63 110


 744 18 232 168
 731 54 202 257 


0
234 117 
 711
(Note that you don't have to calculate these two products from scratch because you already have BC and
AB from the work in Exercise 4.)
David W. Sabo (2000)
Matrix Multiplication
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