matrices1 (1).pptx
TRANSCRIPT
An Overview of Matrix Algebra
Matrix algebra
Matrices
Addition
Multiplication
Translation
Rotation
Scaling
S=T+X
or
S=TX
1
3
2
4
Matrices
Geometric Concept
Matrix Algebra
Definitions
Operations
3
Definition - scalar
a scalar is a real number
Ex: 1,3, 5.5,
4
Matrix
A matrix is an array of numbers
Denoted by Capital letters
All matrices have an order (or dimension):
that is, the number of rows the number of columns. So, A is m by n or (m n).
Example 01: Find the dimensions of each matrix.
Dimension: 3x2
Dimension: 4x1
Dimension: 2x4
Definitions
A square matrix - a matrix that has the same number of rows and columns (n n).
Column Matrix - a matrix that has only one column.
Row Matrix - a matrix that has only one row.
7
Matrix Equality
Two matrices are equal if and only if
they both have the same number of rows and the same number of columns
their corresponding elements are equal
8
* Since the matrices are equal, the corresponding elements are equal!
* Form two linear equations.
Example 02: Find the values for x and y.
Answer
Example 03: Find the values for x and y
Answer
Matrix Operations
Transposition
Addition and Subtraction
Scalar Multiplication
Matrix Multiplication
Inversion
13
The Transpose of a Matrix: AT
The transpose of a matrix is a new matrix that is formed by interchanging the rows and columns of the original matrix.
The transpose of A is denoted by AT
14
Example of a transpose
Thus,
If A = AT, then A is symmetric
15
Addition and Subtraction
Two matrices can be added (or subtracted) iff they are in the same order.
Simply add (or subtract) the corresponding elements. So, A + B = C yields.
16
Addition and Subtraction (cont.)
17
Example 04: (Matrix addition)
Scalar Multiplication
To multiply a scalar times a matrix, simply multiply each element of the matrix by the scalar quantity.
19
Matrix subtraction:
Example 04: (Scalar multiplication and matrix subtraction)
Find (a) 3A (b) B (c) 3A B
(a)
(b)
(c)
Solution:
Matrix Multiplication
In order to multiply matrices, number of columns in the first matrix must equal number of rows in the second matrix.
22
Matrix Multiplication (cont.)
A B = AB
m x n
n x p
m x p
Can you multiply? What will the dimensions be?
2 x 3
3 x 4
AB
5 x 3
5 x 2
2 x 4
Not possible
Ex.
A
B
A
B
AB
How to multiply...
ac
=
ad
2 x 1
1 x 2
2 x 2
How to multiply...
ac
=
ad
2 x 1
1 x 2
2 x 2
bc
bd
Matrix Multiplication (cont.)
Thus
where
26
Example 05
-1(4)
+5(6)
5(-3)
+2(8)
5(4)
+2(6)
0(4)
+(-4)6
0(-3)
+(-4)8
-1(-3)
+5(8)
-4
+ 30
-15
+ 16
20
+12
0
-24
0
-32
3
+ 40
26
1
32
-24
-32
43
=
Dimensions: 2 x 3 2 x 2
Multiplication is not defined
Special matrices
Diagonal
Null
Identity
29
Diagonal Matrix
A diagonal matrix is a square matrix that has values on the diagonal with all off-diagonal entities being zero.
30
Identity Matrix
An identity matrix is a diagonal matrix where all diagonal elements equal one.
I =
A I = A= I A
31
Null Matrix (Zero matrix)
A square matrix where all elements equal zero.
32
Then (1) A+B = B + A
(2) A + ( B + C ) = ( A + B ) + C
(3) ( cd ) A = c ( dA )
(4) 1A = A
(5) c( A+B ) = cA + cB
(6) ( c+d ) A = cA + dA
Properties of matrix addition and scalar multiplication:
Properties
Note:
0mn: the additive identity for the set of all mn matrices
A: the additive inverse of A
Properties of zero matrices:
Properties of transposes:
is a skew-symmetric, find a, b, c?
Example 06:
Sol:
Show that is symmetric
Poof:
Example 07:
ab = ba
(Commutative law for multiplication)
(Sizes are not the same)
(Sizes are the same, but matrices are not equal)
Real number:
Matrix:
Three situations:
Properties
AB does not necessarily equal BA
(BA may even be an impossible operation)
For example,
A B = C
(2 3) (3 2) = (2 2)
B A = D
(3 2) (2 3) = (3 3)
Matrix multiplication is Associative
A(BC) = (AB)C
40
The Determinant of a Matrix
The determinant of a matrix A is denoted by |A| (or det(A)).
Determinants exist only for square matrices.
41
The Determinant of a 2x2 matrix
If A =
Then
42
Minor and Cofactor of a Matrix Entry
Definition:
If A is a square matrix, then the minor of entry aij is denoted by Mij and is defined to be the determinant of the sub-matrix that remains after the ith row and jth column are deleted from A. The number (-1)i+jMij is denoted by Cij and is called the cofactor of entry aij.
For 3x3 matrix
a11 a12 a13
A=
a21
a22 a23
a31
a32 a33
M11 =
a22 a23
a32 a33
C11 =(-1)1+1 M11 = M11
1 2 3
2 5 3
1 0 8
A=
Example
The minor of a11 is:
The cofactor of a11 is:
1 2
1 0
M23=
= 1X0 - 2X1=-2
C23=(-1)i+jMij=(-1)5M23 =-M23 =2
The minor of a23 is:
The cofactor of a23 is:
5 3
0 8
M11=
= 5X8 - 3X0=40
C11=(-1)i+jMij=(-1)2M11=M11=40
a22 a23
a32 a33
C11=
a21 a23
a31 a33
C12= -
a21 a22
a31 a32
C13=
a11 a12 a13
a21 a22 a23
a31 a32 a33
= a11C11 +a12C12 +a13C13
a11 a12 a13
a21 a22 a23
a31 a32 a33
And for nxn matrix: = a11C11 +a12C12 +..+a1nC1n
Finding determinant using the cofactor
Example :
1
3
-1
0
0
4
5
1
2
0
2
1
-3
1
-2
3
A=
det(A)=(1)
4 0 1
5 2 -2
1 1 3
- (0)
3 0 1
-1 2 -2
0 1 3
+ 2
3 4 1
-1 5 -2
0 1 3
- (-3)
3 4 0
-1 5 2
0 1 1
= (1)(35)-0+(2)(62)-(-3)(13)=198
det(A) = a11C11 +a12C12 + a13C13 +a14C14
You can use any row or column to get the value of the determinant
Properties of Determinants
Determinants have several mathematical properties which are useful in matrix manipulations.
1|A|=|AT|.
2.If a row or column of A = 0, then |A|= 0.
3.If every value of a row or column in a matrix A is multiplied by k, then |new matrix| = k|A|.
4.If two rows (or columns) are interchanged the sign, but not value, of |A| changes.
5.If two rows or columns are identical, |A| = 0.
6. If two rows or columns are linear combination of each other, |A| = 0
47
Properties of Determinants
7. |A| remains unchanged if each element of a row or column is multiplied by a constant and added to any other row.
8. |AB| = |A| |B|
9. Det of a diagonal matrix or upper triangular (lower triangular) = product of the diagonal elements
48
The Inverse of a Matrix A (A-1)
For an n n matrix A, there may be a Matrix B such that
AB = I = BA.
The inverse is analogous to a reciprocal
A matrix which has an inverse is nonsingular.
A matrix which does not have an inverse is singular.
An inverse exists only if
Note:
(1) The inverse of A is denoted by
49
50/61
If B and C are both inverses of the matrix A, then B = C.
Pf:
Consequently, the inverse of a matrix is unique.
Theorem : (The inverse of a matrix is unique)
How to find inverse matrixes?
If and |A| 0
51
Properties of inverse matrices
52
The Inverse of a Matrix
The transpose of the matrix of cofactors from A is called the adjoint of A, and it is denoted by Adj(A).
53
How to Find A-1 Using the Adjoint of a Matrix?
If A is an invertible matrix, then
Note:
The square matrix A is invertible if and only if det(A) is not zero.
If A is an n x n triangular matrix, then det(A) is the product of the entries on the main diagonal of the matrix.
54
55/61
Find the inverse of a matrix by Gauss-Jordan Elimination:
Elementary Row Operations
Interchange any two rows.
Multiply a row by a nonzero constant.
Replace a row with itself and a multiple of another row.
56
Solution:
Example : Find the inverse of the following matrix.
58/61
Ex : (Find the inverse of the following matrix)
59/61
Solution
60/61
61/61
62/61
63/61
So the matrix A is invertible, and its inverse is
Check to verify :
64/61
Sol:
Ex : Find the inverse of the following matrix
65/61
66/61
67/61
Power of a square matrix:
Rank
The rank of a matrix is defined as
rank(A) = number of linearly independent rows
= the number of linearly independent columns.
A set of vectors is said to be linearly dependent if scalars c1, c2, , cn (not all zero) can be found such that
c1a1 + c2a2 + + cnan = 0
68
If you have a system of equations and just pick off the coefficients and put them in a matrix it is called a coefficient matrix.
Linear System
Coefficient matrix
If you take the coefficient matrix and then add a last column with the constants, it is called the augmented matrix. Often the constants are separated with a line.
Augmented matrix
"The Goal"
After we get the matrix to look like our goal, we put the variables back in and use back substitution to get the solutions.
We use elementary row operations to make the matrix look like the one below. The # signs just mean there can be any number here---we dont care what.
Row Echelon Form
This is one example only. You can find so many row echelon forms like one below.
Recall
Use row operations to obtain echelon form:
The augmented matrix
Work on this column first. Get the 1 and then use it as a tool to get zeros below it with row operations.
We already have 1 where we need it.
Well take row 1 and multiply it by 3 and add to row 2 to get a 0. The notation for this step is r2 +(3)r1 we write it by the row we replace in the matrix.
r2= r2+(-3)r1
3r1
3 6 3 3
+ r2
3 5 1 3
0 1 2 0
Now well use 2 times row 1 added to row 3 to get a 0 there.
Now our first column is like our goal.
2r1
2 4 2 2
+ r3
2 6 7 1
0 2 5 1
r3= r3 + (-2)r1
r2=(-1)r2
We need a 1 in the second row second column so well multiply row 2 by 1
r3= r3 + (2)r2
2r2
Well use row 2 with the 1 as a tool to get a 0 below it by multiplying it by 2 and adding to row 3
the second column is like we need it now
0 2 4 0
+ r3
0 2 5 1
0 0 1 1
Now well move to the third column and we see for our goal we just need a 1 in the third row of the third column. We have it so weve achieved the goal and its time for back substitution. We put the variables and = signs back in.
Substitute 1 in for z in second equation to find y
x column
y column
z column
equal signs
Substitute 1 in for z and 2 for y in first equation to find x.
Solution is: (2 , 2 , 1)
Solution is: (2 , 2 , 1)
This is the only (x , y , z) that make ALL THREE equations true. Lets check it.
Solution is: (2 , 2 , 1)
Geometrically this means we have three planes that intersect at a point, a unique solution.
The process of reducing the augmented matrix to echelon form or reduced echelon form, and the process of manipulating the equations to eliminate variables, is called:
Gaussian Elimination
Lets try another one:
If you put variables and = signs back in the bottom equation then 0 = 19 a false statement!
INCONSISTENT - NO SOLUTION
Echelon form
Example:
Infinitely many solutions where t is any real number
put variables back in
solve for x & y
Echelon form
Lets try another one:
Let z = t t is any real number
then y = -9-7t
x=(4/5)(-9-7t)-(2/5)t+(6/5)=(-6)t+(-6)
Corresponding equations are
Let s take two parameters called s and t
Few techniques to find the solutions of a linear system if the linear system has a unique solution
Cramers Rule
If a system of n linear equations in n variables has a coefficient matrix with a nonzero determinant,then the solution of the system is given bywhere the ith column of Ai is the column of constants in the system of equations.
3-100
Example
Use Cramers Rule to solve the system of linear equation
Sol:
[
]
ij
a
mn
m
n
n
a
a
a
a
a
a
,
,
,
,
,
,
1
2
21
1
11
=
=
A
1
.
A
=
2
-
1
0
5
-
4
8
2
.
B
=
1
2
3
4
0531
3. C =
2096
-
-
1.
2
x
2
x
+
3
y
=
y
12
3
2
3
=
=
y
x
2
x
=
y
2
x
+
3
y
=
12
2
.
3
x
+
y
x
-
2
y
=
x
+
3
y
-
2
3
x
+
y
=
x
+
3
x
-
2
y
=
y
-
2
1
1
=
=
y
x
=
32
31
22
21
12
11
a
a
a
a
a
a
A
=
32
22
12
31
21
11
a
a
a
a
a
a
T
A
+
+
+
+
+
+
=
+
32
32
31
31
22
22
21
21
12
12
11
11
32
31
22
21
12
11
32
31
22
21
12
11
b
a
b
a
b
a
b
a
b
a
b
a
b
b
b
b
b
b
a
a
a
a
a
a
n
m
ij
n
m
ij
b
B
a
A
=
=
]
[
,
]
[
If
n
m
ij
ij
n
m
ij
n
m
ij
b
a
b
a
B
A
+
=
+
=
+
]
[
]
[
]
[
Then
-
=
+
-
+
+
-
=
-
+
-
3
1
5
0
2
1
1
0
3
2
1
1
2
1
3
1
1
0
2
1
=
-
+
-
-
2
3
1
2
3
1
+
-
+
-
-
2
2
3
3
1
1
=
0
0
0
=
22
21
12
11
22
21
12
11
ka
ka
ka
ka
a
a
a
a
k
scalar
:
,
]
[
If
k
a
A
n
m
ij
=
n
m
ij
ka
kA
=
]
[
Then
-
-
=
2
1
2
1
0
3
4
2
1
A
-
-
=
2
3
1
3
4
1
0
0
2
B
B
A
B
A
)
1
(
-
+
=
-
-
-
=
2
1
2
1
0
3
4
2
1
3
3
A
(
)
-
-
-
=
-
2
3
1
3
4
1
0
0
2
1
B
-
-
-
-
-
=
-
2
3
1
3
4
1
0
0
2
6
3
6
3
0
9
12
6
3
3
B
A
-
-
=
6
3
6
3
0
9
12
6
3
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
-
-
=
2
3
1
3
2
3
1
3
0
3
3
3
4
3
2
3
1
3
-
-
-
-
-
=
2
3
1
3
4
1
0
0
2
-
-
=
4
0
7
6
4
10
12
6
1
b
a
c
d
b
a
=
32
31
22
21
12
11
32
31
22
21
12
11
33
32
31
23
22
21
13
12
11
c
c
c
c
c
c
b
b
b
b
b
b
x
a
a
a
a
a
a
a
a
a
32
33
22
32
12
31
32
31
33
21
32
11
31
31
32
23
22
22
12
21
22
31
23
21
22
11
21
21
32
13
22
12
12
11
12
31
13
21
12
11
11
11
b
a
b
a
b
a
c
b
a
b
a
b
a
c
b
a
b
a
b
a
c
b
a
b
a
b
a
c
b
a
b
a
b
a
c
b
a
b
a
b
a
c
+
+
=
+
+
=
+
+
=
+
+
=
+
+
=
+
+
=
-
-
4
0
2
5
5
1
-
8
6
3
4
39221
.
57634
--
2
1211
1327
2618
x
y
z
-
=
3.
x
+
2
y
-
z
=
1
x
+
3
y
+
2
z
=
7
2
x
+
6
y
+
z
=
8
44
33
22
11
0
0
0
0
0
0
0
0
0
0
0
0
a
a
a
a
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
scalars
:
,
,
,
,
If
d
c
M
C
B
A
n
m
calar
c
M
A
n
m
s
:
,
If
A
A
n
m
=
+
0
(1)
Then
n
m
A
A
=
-
+
0
)
(
(2)
n
m
n
m
or A
c
cA
=
=
=
0
0
0
)
3
(
A
A
T
T
=
)
(
)
1
(
T
T
T
B
A
B
A
+
=
+
)
(
)
2
(
)
(
)
(
)
3
(
T
T
A
c
cA
=
)
(
)
4
(
T
T
T
A
B
AB
=
=
0
3
0
2
1
0
If
c
b
a
A
=
0
3
0
2
1
0
c
b
a
A
-
-
-
-
-
-
=
-
0
3
2
0
1
0
c
b
a
A
T
T
A
A
-
=
3
,
2
,
1
-
=
-
=
-
=
c
b
a
symmetric
is
)
(
)
(
T
T
T
T
T
T
T
AA
AA
A
A
AA
\
=
=
T
AA
undefined.
is
,
defined
is
then
,
If
BA
AB
p
m
(1)
m
m
m
m
M
BA
M
AB
n
p
m
=
=
(3)
,
then
,
If
n
n
m
m
M
BA
M
AB
n
m
p
m
=
(2)
,
then
,
,
If
BA
AB
p
n
n
m
21
12
22
11
a
a
a
a
A
-
=
22
21
12
11
a
a
a
a
0
A
1
-
A
I
A
A
AA
=
=
-
-
1
1
)
2
(
C
B
C
IB
C
B
CA
CI
AB
C
I
AB
=
=
=
=
=
)
(
)
(
-
-
=
a
c
b
d
-
)
det(
1
1
A
A
=
d
c
b
a
A
(
)
1
1
1
-
-
A
B
AB
=
-
(
)
(
)
T
-
1
1
A
A
T
=
-
(
)
A
A
=
-
1
1
-
=
33
32
31
23
22
21
13
12
11
c
c
c
c
c
c
c
c
c
Matrix
Cofactor
=
=
33
23
13
32
22
12
31
21
11
)
(
c
c
c
c
c
c
c
c
c
C
A
Adj
T
A
-
1
=
1
d
e
t
(
A
)
A
d
j
(
A
)
[
]
[
]
1
n
Eliminatio
Jordan
-
Gauss
|
|
-
A
I
I
A
-
-
-
=
3
2
6
1
0
1
0
1
1
A
1
0
0
3
2
6
0
1
1
1
1
0
0
0
1
0
1
1
1
2
2
)
1
(
'
-
-
-
-
-
+
=
M
M
M
r
r
r
[
]
-
-
-
=
1
0
0
0
1
0
0
0
1
3
2
6
1
0
1
0
1
1
M
M
M
M
I
A
(
)
-
-
-
-
+
=
1
0
6
0
1
1
0
0
1
3
4
0
1
1
0
0
1
1
1
3
3
6
'
M
M
M
r
r
r
-
-
-
-
+
=
1
4
2
1
0
0
0
1
1
1
1
0
0
0
1
0
1
1
2
3
3
4
'
M
M
M
r
r
r
-
-
-
-
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