Download PowerPoint

159 Lecture 19
Elementary Matrix Theory
Matrix Definition
 A matrix is a
rectangular array
of elements
(usually numbers)
written in rows and
columns.
 Example 1: Some
matrices:
2
Matrix Definition
 Example 1 (cont.):






Matrix A is a 3 x 2
matrix of integers.
A has 3 rows and 2
columns.
Matrix B is a 2 x 2
matrix of rational
numbers.
Matrix C is a 1 x 4
matrix of real numbers.
We also call C a row
vector.
A matrix consisting of a
single column is often
called a column
vector.
3
Matrix Definition
 Notation:
4
Arithmetic with Matrices
 Matrices of the same size (i.e. same
number of rows and same number of
columns), with elements from the
same set, can be added or
subtracted!
 The way to do this is to add or
subtract corresponding entries!
5
Arithmetic with Matrices
6
Arithmetic with Matrices
 Example 2: For matrices A and B
given below, find A+B and A-B.
7
Arithmetic with Matrices
 Example 2 (cont):
Solution:
 Note that A+B and A-B are the same
size as A and B, namely 2 x 3.
8
Arithmetic with Matrices
 Matrices can also be multiplied. For AB to make sense,
the number of columns in A must equal the number of
rows in B.
9
Arithmetic with Matrices
 Example 3: For matrices A and B
given below, find AB and BA.
10
Arithmetic with Matrices


Example 3 (cont.):
A x B is a 3 x 2 matrix. To get the row i, column j entry of this matrix,
multiply corresponding entries of row i of A with column j of B and add.

Since B has 2 columns and A has 3 rows, we cannot find the product
BA (# columns of 1st matrix must equal # rows of 2cd matrix).
11
Arithmetic with Matrices
 Another useful operation with
matrices is scalar multiplication,
i.e. multiplying a matrix by a number.
 For scalar k and matrix A, kA=Ak is
the matrix formed by multiplying
every entry of A by k.
12
Arithmetic with Matrices
 Example 4:
13
Identities and Inverses
 Recall that for any real number a,
a+0 = 0+a = a and (a)(1) = (1)(a) = a.
 We call 0 the additive identity and 1 the
multiplicative identity for the set of real
numbers.
 For any real number a, there exists a real
number –a, such that
a+(-a) = -a+a = 0.
 Also, for any non-zero real number a, there
exists a real number a-1 = 1/a, such that
(a-1)(a) = (a)(a-1) = 1.
 We all –a and a-1 the additive inverse and
multiplicative inverse of a, respectively.
14
Identities and Inverses
 For matrices, we also have an additive identity
and multiplicative identity!
15
Identities and Inverses
A+0 = 0+A = A and AI = IA = A holds.
(HW-check!)
16
Identities and Inverses
 Clearly, A+(-A) = -A + A = 0 follows! Note also that B-A
= B+(-A) holds for any m x n matrices A and B.
17
Identities and Inverses
 Example 5:
18
Identities and Inverses
 Example 5 (cont):
19
Identities and Inverses
 Example 5 (cont.)
20
Identities and Inverses
 Example 5 (cont.)
21
Identities and Inverses
 Example 5 (cont):
22
Identities and Inverses
 For multiplicative inverses, more work
is needed.
 For example, here is one way to find
the matrix A-1, given matrix A, in the
2 x 2 case!
23
Identities and Inverses
24
Identities and Inverses





From the first matrix equation, we see that e, f, g, and h
must satisfy the system of equations:
ae + bg = 1
af + bh = 0
ce + dg = 0
cf + dh = 1.
It follows that if e, f, g, and h satisfy this system, then the
second matrix equation above also holds!
Solving the system of equations, we find that ad-bc  0
must hold and
e = d/(ad-bc),
f = -b/(ad-bc),
g = -c/(ad-bc),
h = a/(ad-bc).
Thus, we have the following result for 2 x 2 matrices:
25
Identities and Inverses
 In this case, we say A is invertible.
 If ad-bc = 0, A-1 does not exist and
we say A is not invertible.
 We call the quantity ad-bc the
determinant of matrix A.
26
Identities and Inverses
 Example 6: For matrices A and B
below, find A-1 and B-1, if possible.
27
Identities and Inverses
 Example 6 (cont.)
 Solution: For matrix A, ad-bc = (1)(4)-(2)(3)= 4-6 =
-2 0, so A is invertible. For matrix B, ad-bc = (3)(2)(1)(6) = 6-6 = 0, so B is not invertible.
 HW-Check that AA-1 = A-1A = I!!
 Note: For any n x n matrix, A-1 exists, provided the
determinant of A is non-zero.
28
Linear Systems of Equations
 One use of matrices is to solve
systems of linear equations.
 Example 7: Solve the system
x + 2y = 1
3x + 4y = -1
 Solution: This system can be written
in matrix form AX=b with:
29
Linear Systems of Equations
 Example 7 (cont.)
 Since we know from Example 6 that
A-1 exists, we can multiply both sides
of AX = b by A-1 on the left to get:
A-1AX = A-1b => X = A-1b.
 Thus, we get in this case:
30
Linear Systems of Equations
 Example 7 (cont.):
31
References
 Elementary Linear Algebra (4th ed) by
Howard Anton.
 Cryptological Mathematics by Robert
Edward Lewand (section on
matrices).
32