Transcript 2.1
MAT 2401
Linear 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
1
2
m
2
n
Notations
Matrix
A aij
j th
i th
aij
Notations
Matrix
Example:
A aij
1 1 1 4
A 2 2 5 11
4 6 8 24
j th
i th
aij
a11
a23
Special Cases
Row Vector
b1
b2
bn
Column Vector
c1
c
2
cm
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
A
3
1
0 2
B
3
2
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
A
3 1
2A
Matrix Multiplication
Define multiplications between 2
matrices
Not “term-by-term” operations
Motivation
The LHS of the linear equation consists of
two pieces of information:
• coefficients: 2, -3, and 4
• variables: x, y, and z
2x 3y 4z 5
Motivation
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.
2x 3y 4z 5
x
2 3 4 y 5
z
Row-Column Product
a1
a2
b1
b
an 2 a1b1 a2b2
bn
same no. of elements
anbn
Example 3
2
2
1 3 2 4 1
2
Matrix Multiplication
a11
ai1
am1
a12
ai 2
am 2
a1 p
b11
b
aip 21
b
amp p1
ai1b1 j ai 2b2 j
b1 j
b2 j
bpj
aip bpj
b1n
b2 n
bpn
j th
i th
cij
Example 4
1 2 0 1
1 0 1 0
Example 5 (a)
4 2
1 2 1
0
1
2 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)
1
1 1 2
Scratch:
Q: Is it possible to multiply the
2 matrices?
Q: What is the dimension of the
resulting matrix?
Example 5 (d)
1
1 2 1
Scratch:
Q: Is it possible to multiply the
2 matrices?
Q: What is the dimension of the
resulting matrix?
1
Remark: A 1 2 , B
1
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?
1 1
1 1
, B
Remark: A
1
1
1
1
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?
1 0
1 2
, A
Remark: I
0
1
3
4
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
I In
0
0
1
0
0
0
1
Zero Matrix
mxn Matrix with all zero entries
0 0mn
0
0
0
0
0
0
0
0
0
Representation of Linear
System by Matrix Multiplication
x yz 4
2 x 2 y 5 z 11
4 x 6 y 8 z 24
Representation of Linear
System by Matrix Multiplication
x yz 4
2 x 2 y 5 z 11
4 x 6 y 8 z 24
Representation of Linear
System by Matrix Multiplication
x yz 4
2 x 2 y 5 z 11
4 x 6 y 8 z 24
Representation of Linear
System by Matrix Multiplication
Let
x yz 4
2 x 2 y 5 z 11
4 x 6 y 8 z 24
1 1 1
x
4
A 2 2 5 , X y , b 11
4 6 8
z
24
Then the linear system is given by
Remark
It would be nice
if “division” can
be defined such
that:
Let
1 1 1
x
4
A 2 2 5 , X y , b 11
4 6 8
z
24
Then the linear system is given by
(2.3) Inverse
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.