Download Matrix Determinants

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.
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 
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
4
2 1  3 3  11
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
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
5
2
2
1 4 2
3
3
60 16
6 5
4 3
2
1
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
5
2
2
1 4 2
3
3
60
6 5
4 3
16
2
1
= (-8) - (94) = -102
3
-20 -24
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