Transcript Inverse Matrices

MATRICES
MATRIX OPERATIONS
About Matrices
 A matrix is a rectangular
arrangement of numbers in rows and
columns. Rows run horizontally and
columns run vertically.
 The dimensions of a matrix are stated
“m x n” where ‘m’ is the number of
rows and ‘n’ is the number of
columns.
Equal Matrices
 Two matrices are considered equal
if they have the same number of
rows and columns (the same
dimensions) AND all their
corresponding elements are exactly
the same.
Special Matrices
Some matrices have special names
because of what they look like.
a) Row matrix: only has 1 row.
b) Column matrix: only has 1
column.
c) Square matrix: has the same
number of rows and columns.
d) Zero matrix: contains all zeros.
Matrix Addition
 You can add or subtract matrices if
they have the same dimensions
(same number of rows and
columns).
 To do this, you add (or subtract)
the corresponding numbers
(numbers in the same positions).
Matrix Addition
Example:
 2 4  1 0 
 5 0    2 1  

 

1 3  3 3
 3 4 
 7 1


 2 0 
Real-Life Example
Let’s say you’re in avid reader, and in June, July, and August you read fiction and
non-fiction books, and magazines, both in paper copies and online. You want to
keep track of how many different types of books and magazines you read, and store
that information in matrices. Here is that information, and how it would look in
matrix form:
Solution

We can add matrices if the dimensions are the same; since
the three matrices are all “3 by 2”, we can add them. For
example, if we wanted to know the total number of each type of
book/magazine we read, we could add each of the elements to
get the sum:
Scalar Multiplication
 To do this, multiply each entry in
the matrix by the number outside
(called the scalar). This is like
distributing a number to a
polynomial.
Scalar Multiplication
Example:
 2 4  8 16 




4 5 0   20 0


1 3  4 12 
Matrix Multiplication


Matrix Multiplication is NOT
Commutative! Order matters!
You can multiply matrices only if the
number of columns in the first matrix
equals the number of rows in the second
matrix.
2 columns
3
2
 5 6    1

  3
 9 7 
2
4
0
5 
2 rows
Matrix Multiplication

Take the numbers in the first row of
matrix #1. Multiply each number by its
corresponding number in the first
column of matrix #2. Total these
products.
3
2
 5 6    1

  3
 9 7 
2(1)+3(3)=11
2
4
0
5 
The result, 11, goes in
row 1, column 1 of the
answer. Repeat with
row 1, column 2; row 1
column 3; row 2,
column 1; ...
Matrix Multiplication

Notice the dimensions of the matrices and
their product.
3
2
 5 6    1

  3
 9 7 
3x2
__
8 15 
 11
2 0  


13
34
30

4 5  
 12 46 35 
2 x__
3
3 x__
3
__
Matrix Multiplication

Another example:
2 1
 9 0    5  

  2 
10 5  
3x2
2x1
 8 
 45


 60 
3x1
Real-World Application

Let’s say we want to find the final grades for 3 girls, and we know what
their averages are for tests, projects, homework, and quizzes. We also
know that tests are 40% of the grade, projects 15%, homework 25%,
and quizzes 20%.
Solution
So Alexandra has a 90, Megan has a 77, and Brittney has an 87.
Matrix Determinants
 A Determinant is a real number associated
with a matrix. Only SQUARE matrices
have a determinant.
 The symbol for a determinant can be the
phrase “det” in front of a matrix variable,
det(A); or vertical bars around
a matrix, |A| or 3 1 .
2
4
Matrix Determinants
To find the determinant of a 2 x 2 matrix,
multiply diagonal #1 and subtract the product
of diagonal #2.
Diagonal 2 = -2
3 1

2 4
Diagonal 1 = 12
12  (2)  14
Matrix Determinants
To find the determinant of a 3 x 3 matrix, first
recopy the first two columns. Then do 6
diagonal products.
18
60 16
5 2 6 5 2
2 1 4 2 1
3 3 4 3 3
-20 -24
36
Matrix Determinants
The determinant of the matrix is the sum of
the downwards products minus the sum of the
upwards products.
18
60
16
5 2 6 5 2
2 1 4 2 1
3 3 4 3 3
-20 -24
= (-8) - (94) = -102
36
Identity Matrices

An identity matrix is a square matrix that
has 1’s along the main diagonal and 0’s
everywhere else.
1 0 0 
0 1 0 


 0 0 1 

1 0 
0 1 


When you multiply a matrix by the
identity matrix, you get the original
matrix.
Inverse Matrices
 When you multiply a matrix and its
inverse, you get the identity matrix.
 3 1  2 1   1 0 
 5 2   5 3   0 1 


 

Inverse Matrices
 Not all matrices have an inverse!
 To find the inverse of a 2 x 2 matrix,
first find the determinant.
a) If the determinant = 0, the inverse does
not exist!
 The inverse of a 2 x 2 matrix is the
reciprocal of the determinant times the
matrix with the main diagonal swapped
and the other terms multiplied by -1.
Inverse Matrices
 3 1
Example 1: A  

5
2


det(A)  6  (5)  1
1 2 1 2 1
A  


1  5 3  5 3
1
Inverse Matrices
Example 2:
 2 2 
B

5
4


det(B)  (8)  (10)  2
2 2
1 4
B  
 5

2  5 2    2
1
1
1