mat 2401 linear algebra
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MAT 2401Linear Algebra
2.1 Operations with Matrices
http://myhome.spu.edu/lauw
HW...
If you do not get 9 points or above on #1, you are not doing the GJE correctly. Some of you are doing RE.
GJE is the corner stone of this class, you really need to figure it out.
Today
Written HW Again, today may be longer. It is
more efficient to bundle together some materials from 2.2.
Next class session will be shorter.
Preview
Look at the algebraic operations of matrices
“term-by-term” operations•Matrix Addition and Subtraction
•Scalar Multiplication Non-“term-by-term” operations
•Matrix Multiplication
Matrix
If a matrix has m rows and n columns, then the size (dimension) of the matrix is said to be mxn.
1 2
1
2
n
m
Notations
Matrix
th
t
h
ij
ij
j
A a
ai
Notations
Matrix Example:
11
23
1 1 1 4
2 2 5 11
4 6 8 24
A
a
a
th
t
h
ij
ij
j
A a
ai
Special Cases
Row Vector
Column Vector
1 2 nb b b
1
2
m
c
c
c
Matrix Addition and Subtraction
Let A = [aij] and B = [bij] be mxn matrices
Sum: A + B = [aij+bij]
Difference: A-B = [aij-bij]
(Term-by term operations)
Example 1
1 2
3 1
0 2
3 2
A
B
A B
A B
Scalar Multiplication
Let A = [aij] be a mxn matrix and c a scalar.
Scalar Product: cA=[caij]
Example 2
1 2
3 1A
2A
Matrix Multiplication
Define multiplications between 2 matrices
Not “term-by-term” operations
Motivation
2 3 4 5x y z
The LHS of the linear equation consists of two pieces of information:•coefficients: 2, -3, and 4
•variables: x, y, and z
Motivation
2 3 4 5
2 3 4 5
x y z
x
y
z
Since both the coefficients and variables can be represented by vectors with the same “length”, it make sense to consider the LHS as a “product” of the corresponding vectors.
Row-Column Product
1
21 2 1 1 2 2n n n
n
b
ba a a a b a b a b
b
same no. of elements
Example 3
2
21 3 2 4
1
2
Matrix Multiplication
1
21
11 12 111 1
21 22
2
2
1
1 1 2
1
j
ji i ip
pj
i j i j
pn
n
p pnm m mp
ip pj
b
ba a a
b
a b a
a a ab b
b b
b b
b a
a
b
a a
th
th ijc
j
i
Example 4
1 2 0 1
1 0 1 0
Example 5 (a)
4 21 2 1
0 12 3 1
2 1
Scratch:Q: Is it possible to multiply the 2 matrices?
Q: What is the dimension of the resulting matrix?
Example 5 (b)
1 2 3 2
2 3 1 3
Scratch:Q: Is it possible to multiply the 2 matrices?
Q: What is the dimension of the resulting matrix?
Example 5 (c)
11 2
1
Scratch:Q: Is it possible to multiply the 2 matrices?
Q: What is the dimension of the resulting matrix?
Example 5 (d)
11 2
1
Scratch:Q: Is it possible to multiply the 2 matrices?
Q: What is the dimension of the resulting matrix?
Remark: 11 2 ,
1A B
Example 5 (e)
1 1 1 1
1 1 1 1
Scratch:Q: Is it possible to multiply the 2 matrices?
Q: What is the dimension of the resulting matrix?
Remark:1 1 1 1
, 1 1 1 1
A B
Example 5 (f)
1 0 1 2
0 1 3 4
Scratch:Q: Is it possible to multiply the 2 matrices?
Q: What is the dimension of the resulting matrix?
Remark:1 0 1 2
, 0 1 3 4
I A
Interesting Facts
The product of mxp and pxn matrices is a mxn matrix.
In general, AB and BA are not the same even if both products are defined.
AB=0 does not necessary imply A=0 or B=0.
Square matrix with 1 in the diagonal and 0 elsewhere behaves like multiplicative identity.
Identity Matrix
nxn Square Matrix
1 0 0
0 1
0
0 0 1
nI I
Zero Matrix
mxn Matrix with all zero entries
0 0 0
0 00 0
0
0 0 0
mn
Representation of Linear System by Matrix Multiplication
4
2 2 5 11
4 6 8 24
x y z
x y z
x y z
Representation of Linear System by Matrix Multiplication
2 2 5 11
4
4
4 6 8 2
x y z
z
x y z
x y
Representation of Linear System by Matrix Multiplication
4
4 6 8
11
2
2 5
4
2x y
y
z
x z
x y z
Let
Then the linear system is given by
Representation of Linear System by Matrix Multiplication
4
2 2 5 11
4 6 8 24
x y z
x y z
x y z
1 1 1 4
2 2 5 , , 11
4 6 8 24
x
A X y b
z
Let
Then the linear system is given by
Remark
It would be nice if “division” can be defined such that:
(2.3) Inverse
1 1 1 4
2 2 5 , , 11
4 6 8 24
x
A X y b
z
HW...
If you do not get 9 points or above on #1, you are not doing the GJE correctly. Some of you are doing RE.
GJE is the corner stone of this class, you really need to figure it out.
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