This document provides an overview of matrix algebra concepts including: - Matrices are rectangular arrays of numbers with rows and columns. - Matrix operations include addition, subtraction, scalar multiplication, and multiplication. - Special matrices include the zero matrix, identity matrix, and inverse matrices. - The determinant of a square matrix is important for determining if the matrix has an inverse.

1. 7.2
Matrix Algebra
Copyright © 2011 Pearson, Inc.

2. What you’ll learn about
 Matrices
 Matrix Addition and Subtraction
 Matrix Multiplication
 Identity and Inverse Matrices
 Determinant of a Square Matrix
 Applications
… and why
Matrix algebra provides a powerful technique to
manipulate large data sets and solve the related
problems that are modeled by the matrices.
Copyright © 2011 Pearson, Inc. Slide 7.2 - 2

3. Matrix
Let m and n be positive integers. An m  n matrix
(read "m by n matrix") is a rectangular array of
m rows and n columns of real numbers.
a11 a12 L a1n
a21 a22 L a2n
M M M
am1 am2 L amn















We also use the shorthand notation aij

for this matrix.
Copyright © 2011 Pearson, Inc. Slide 7.2 - 3

4. Matrix Vocabulary
Each element, or entry, aij, of the matrix uses
double subscript notation. The row subscript is
the first subscript i, and the column subscript is
j. The element aij is the ith row and the jth
column. In general, the order of an m  n
matrix is m  n.
Copyright © 2011 Pearson, Inc. Slide 7.2 - 4

5. Example Determining the Order of a
Matrix
What is the order of the following matrix?

1 4 5
3 5 6





Copyright © 2011 Pearson, Inc. Slide 7.2 - 5

6. Example Determining the Order of a
Matrix
What is the order of the following matrix?
1 4 5

3 5 6


The matrix has 2 rows and 3 columns
so it has order 2  3.



Copyright © 2011 Pearson, Inc. Slide 7.2 - 6

7. Matrix Addition and Matrix
Subtraction


Let A  aij
and 
B  

 bij

 be matrices of order m n.
1. The sum A+ B is the m n matrix


A B  aij  bij

.
2. The difference A B is the m n matrix


A B  aij  bij

.
Copyright © 2011 Pearson, Inc. Slide 7.2 - 7

8. Example Matrix Addition
1 2 3
4 5 6
Copyright © 2011 Pearson, Inc. Slide 7.2 - 8






2 3 4
5 6 7







9. Example Matrix Addition
1 2 3
4 5 6
A B 
2 1 2  3 3 4
4  5 5 6 6  7







3 5 7
9 11 13





Copyright © 2011 Pearson, Inc. Slide 7.2 - 9






2 3 4
5 6 7







10. Example Using Scalar Multiplication
3
1 2 3
4 5 6






Copyright © 2011 Pearson, Inc. Slide 7.2 - 10

11. Example Using Scalar Multiplication
1 2 3
4 5 6




31 32 33
34 35 36







3 6 9
12 15 18





3



Copyright © 2011 Pearson, Inc. Slide 7.2 - 11

12. The Zero Matrix
The m n matrix 0  [0] consisting entirely of
zeros is the zero matrix.
Copyright © 2011 Pearson, Inc. Slide 7.2 - 12

13. Additive Inverse

Let A  aij

be any m n matrix.
 
The m n matrix B  aij
consisting of the additive
inverses of the entries of A is the additive inverse of A
because A B  0.
Copyright © 2011 Pearson, Inc. Slide 7.2 - 13

14. Matrix Multiplication


Let A  aij
be 
any m r matrix and B  

 bij


be any r  n matrix.
The product AB  cij



 is the m n matrix where
cij  ai1b1 j +ai2b2 j  ...  airbrj .
Copyright © 2011 Pearson, Inc. Slide 7.2 - 14

15. Example Matrix Multiplication
Find the product AB if possible.
A 

1 2 3
0 1 1




 and B 

1 0
2 1
0 1









Copyright © 2011 Pearson, Inc. Slide 7.2 - 15

16. Example Matrix Multiplication
A 

1 2 3
0 1 1




 and B 

1 0
2 1
0 1









The number of columns of A is 3 and the number of
rows of B is 3, so the product is defined.
The product AB  cij


is a 2  2 matrix where
c11  1 2 3 

1
2
0










 11 2  2  30  5,
Copyright © 2011 Pearson, Inc. Slide 7.2 - 16

17. Example Matrix Multiplication
A 

1 2 3
0 1 1




 and B 

1 0
2 1
0 1










 0
c12  1 2 3 




1
1





 10  2 1 3 1  1,
c21  0 1 1 

1
2
0










 011 2  10  2,
Copyright © 2011 Pearson, Inc. Slide 7.2 - 17

18. Example Matrix Multiplication
A 

1 2 3
0 1 1




 and B 

1 0
2 1
0 1









c22  0 1 1  

0
1
1









 0 0 11 1 1  2.
Thus AB 

5 1
2 2




.
Copyright © 2011 Pearson, Inc. Slide 7.2 - 18

19. Identity Matrix
The n  n matrix In with 1's on the main diagonal and
0's elsewhere is the identity matrix of order n  n.
In 
1 0 0 L 0
0 1 0 L 0
0 0 1 L 0
M M M 0
0 0 0 0 1
















Copyright © 2011 Pearson, Inc. Slide 7.2 - 19

20. Inverse of a Square Matrix

Let A  aij

be an n  n matrix.
If there is a matrix B such that
AB  BA  In ,
then B is the inverse of A. We write B  A1.
Copyright © 2011 Pearson, Inc. Slide 7.2 - 20

21. Inverse of a 2 × 2 Matrix
If ad  bc  0, then
a b
c d

 
  1


1
ad  bc
d b
c a

 

 
.
Copyright © 2011 Pearson, Inc. Slide 7.2 - 21

22. Determinant of a Square Matrix


Let A  aij

 be a matrix of order n  n (n  2).
The determinant of A, denoted by det A or | A | ,
is the sum of the entries in any row or any column
multiplied by their respective cofactors. For
example, expanding by the ith row gives
det A | A | ai1Ai1  ai2Ai2  ...  ainAin .
Copyright © 2011 Pearson, Inc. Slide 7.2 - 22

23. Inverses of n  n Matrices
An n  n matrix A has an inverse if and only if
det A ≠ 0.
Copyright © 2011 Pearson, Inc. Slide 7.2 - 23

24. Example Finding Inverse Matrices
Determine whether the matrix has an inverse.
If so, find its inverse matrix.
A 

5 1
8 3





Copyright © 2011 Pearson, Inc. Slide 7.2 - 24

25. Example Finding Inverse Matrices
A 

5 1
8 3





Since det A  ad  bc  5318  7  0,
we conclude that A has an inverse.
Use the formula A1 
1
ad  bc
d b
c a

 

 

1
7

3 1
8 5

 
 

3
7

1
7

8
7
5
7












.
Copyright © 2011 Pearson, Inc. Slide 7.2 - 25

26. Example Finding Inverse Matrices
A 

5 1
8 3



Check:
A1A 


3
7

1
7

8
7
5
7













5 1
8 3





3
7

8
7
3
7

3
7

40
7

40
7

8
7

15
7













1 0
0 1





 I2
Similarly, A1A  I2 .
Copyright © 2011 Pearson, Inc. Slide 7.2 - 26

27. Properties of Matrices
Let A, B, and C be matrices whose orders are such that
the following sums, differences, and products are
defined.
1. Community property
Addition: A + B = B + A
Multiplication: Does not hold in general
2. Associative property
Addition: (A + B) + C = A + (B + C)
Multiplication: (AB)C = A(BC)
3. Identity property
Addition: A + 0 = A
Multiplication: A·In = In·A = A
Copyright © 2011 Pearson, Inc. Slide 7.2 - 27

28. Properties of Matrices
Let A, B, and C be matrices whose orders are such that
the following sums, differences, and products are
defined.
4. Inverse property
Addition: A + (-A) = 0
Multiplication: AA-1 = A-1A = In |A|≠0
5. Distributive property
Multiplication over addition:
A(B + C) = AB + AC (A + B)C = AC + BC
Multiplication over subtraction:
A(B – C) = AB – AC (A – B)C = AC – BC
Copyright © 2011 Pearson, Inc. Slide 7.2 - 28

29. Quick Review
The points (a) (1,  3) and (b) (x, y) are reflected
across the given line.
Find the coordinates of the reflected points.
1. The x-axis
2. The line y  x
3. The line y  x
Expand the expression,
4. sin(x  y)
5. cos(x  y)
Copyright © 2011 Pearson, Inc. Slide 7.2 - 29

30. Quick Review Solutions
The points (a) (1,  3) and (b) (x, y) are reflected
across the given line.
Find the coordinates of the reflected points.
1. The x-axis (a) (1,3) (b) (x,  y)
2. The line y  x (a) (  3,1) (b) ( y, x)
3. The line y  x (a) (  3, 1) (b) ( y, x)
Expand the expression,
4. sin(x  y) sin x cos y  sin y cos x
5. cos(x  y) cos x cos y  sin x sin y
Copyright © 2011 Pearson, Inc. Slide 7.2 - 30