Avon High School
Section: 9.3
ACE COLLEGE ALGEBRA II - NOTES
Matrix Operations and Their Applications
Mr. Record: Room ALC-129
Semester 2 - Day 16
Notations for Matrices
Example 1
Matrix Notation
Let
 5 2 
A   3  
 1
6 
a. What is the order of A?
b. Identify a12 and a31 .
Equality of Matrices
Two matrices are equal if and only if they have the same order and corresponding elements are equal.
Definition of Equality of Matrices
Two matrices A and B are equal if and only if they have the same order m n
and aij  bij for i  1, 2, , m and j  1, 2, n
Example 2
Adding and Subtracting Matrices
Perform the indicated matrix operations:
 4 3  6 3
a. 


 7 6  2 4
 5 4   4 8 
b.  3 7    6 0 

 

 0 1   5 3
Note: You may only add and subtract matrices that are of the same order.
0 0 0 
A matrix whose elements are all 0 is called the zero matrix. E.g. 0 0 0 


0 0 0 
Properties of Matrix Addition
If A, B and C are m n matrices and 0 is the m n zero matrix, then the
following properties are true:
1. A  B  B  A
Commutative Property of Addition
2. ( A  B)  C  A  ( B  C )
Associative Property of Addition
3. A  0  0  A  A
Additive Identity Property
4. A  ( A)  ( A)  A  0
Additive Inverse Property
Scalar Multiplication
Definition of Scalar Multiplication
If A   aij  is a matrix of m n and c is scalar, then the matrix cA is the m n
matrix given by cA  caij  .
The matrix is obtained by multiplying each element of A by the real number c.
We call cA a scalar multiple of A.
Example 3
Scalar Multiplication
 4 1 
 1 2
and B  
If A  

 , find the following matrices:
 3 0
8 5
a. 6B
b. 3A  2B
Properties of Scalar Multiplication
If A and B are m n matrices, and c and d are scalars, then the following
properties are true:
1. (cd ) A  c(dA)
Multiplication
2. 1A  A
3. c( A  B )  cA  cB
4. (c  d ) A  cA  dA
Associative Property of Scalar
Scalar Identity Property
Distributive Property
Distributive Property
Example 4
Solving a Matrix Equation
Solve for X in the matrix equation 3 X  A  B, where
 2 8
 10 1 
A
and B  


 9 17 
0 4 
Matrix Multiplication
Beware: Matrix Multiplication is not performed as you might think.
Definition of Matrix Multiplication
Row 1 of A
x Column 1
of B
a b   e
AB  

c d   g
f   ae  bg af  bh 

h   ce  dg cf  dh 
Row 2 of A
x Column 1
of B
Example 5
l
Row 1 of A
x Column 2
of B
Row 2 of A
x Column 2
of B
Multiplying Matrices
Multiply each pair of matrices where possible.
 1 3  4 6 
a. 


 2 5  1 0 
1 
b.  2 0 4  3 
 7 
1 3   2 3 1 6 
c. 


0 2   0 5 4 1 
 2 3 1 6  1 3 
d. 


 0 5 4 1  0 2 
Properties of Matrix Multiplication
If A, B and C are matrices and c is a scalar, then the following properties are
true. (Assume the order of each matrix is such that all operation listed are
defined).
1. ( AB)C  A( BC )
2. ( A  B)C  AC  BC
A( B  C )  AB  AC
3. c( AB)  (cA) B
Associative Property of Matrix Multiplication
Distributive Properties of Matrix
Associative Property of Scalar Multiplication
Applications
Example 6
Transformations of an Image
0 3 4 
Consider the triangle represented by the matrix A  

0 5 2 
Use matrix operations to perform the following transformations:
a. Move the triangle 3 units to the left and 1 unit down.
b. Enlarge the triangle to twice its original perimeter.
c. Illustrate your results in parts (a) and (b) by showing the original triangle and the transformed image in a
rectangular coordinate system.
1 0 
d. Let B  
 . Find BA. What effect does this have on the original image?
0 1