1 matrices and determinants sjk
TRANSCRIPT
-
8/10/2019 1 Matrices and Determinants Sjk
1/40
Lecture 1
Matrices and Determinants
MO 091204 -Mathematic Engineering
Ocean Engineering ITS
-
8/10/2019 1 Matrices and Determinants Sjk
2/40
1.1 atrices
1.2 Operations of matrices
. ypes o matr ces
1.4 Pro erties of matrices1.5 Determinants
1.6 Inverse of a 33 matrix
-
8/10/2019 1 Matrices and Determinants Sjk
3/40
1.1 Matrices
2 3 7A = 2 1 4B
=
4 7 6
.
rectangular array of numbers enclosed by a pair of bracket.
Why matrix?
-
8/10/2019 1 Matrices and Determinants Sjk
4/40
1.1 Matrices
7,+ =x y
It is easy to show thatx = 3 andy = 4.
3 5. =x y
2 7,+ = x y z
How about solving2 4 2,
5 4 10 1,
=
+ + =
x y z
x y z
3 6 5. = x y z
-
8/10/2019 1 Matrices and Determinants Sjk
5/40
11 12 1 K na a a
1.1 Matrices
21 22 2 =
M On
a a aA
1 2m m mn
numbers ai are called elements. First subscript indicates therow; second subscript indicates the column. The matrix
consists of mn elements
It is called the m n matrixA = [aij] or simply the matrixA if number of rows and columns are understood.
-
8/10/2019 1 Matrices and Determinants Sjk
6/40
S uare matrices
1.1 Matrices
When m = n, i.e.,11 12 1
K na a a
a a a
1 2
=
M O
n
n n nn
A
a a a
A is called a square matrix of order n or n-square
matrix
elements a11, a22, a33,, ann called diagonal elements.
....
11 221
+ + +
=
=
n
ii nni a a a a
-
8/10/2019 1 Matrices and Determinants Sjk
7/40
E ual matrices
1.1 Matrices
Two matricesA = [a ] andB = [b ] are said to be e ual(A =B) if each element ofA is equal to the corresponding
element ofB, i.e., aij = bij for 1 i m, 1 j n.
ifpronouns if and only if
, ij ij , ;if aij = bij for 1 i m, 1 j n, it impliesA =B.
-
8/10/2019 1 Matrices and Determinants Sjk
8/40
E ual matrices
1.1 Matrices
1 0 =
a bB
=
Given thatA =B, find a, b, c and d.
4 2 c d
ifA =B, then a = 1, b = 0, c = -4 and d= 2.
-
8/10/2019 1 Matrices and Determinants Sjk
9/40
Zero matrices
1.1 Matrices
Ever element of a matrix is zero, it is called a zero matrix,
i.e.,
0 0 0 K0 0 0 =
M OA
-
8/10/2019 1 Matrices and Determinants Sjk
10/40
Sums of matrices
. pera ons o ma r ces
IfA = [aij] andB = [bij] are m n matrices, thenA + Bs e ne as a ma r x = , w ere = ij , ij = ij
+ bij for 1 i m, 1 j n.
0 1 4=
A
1 2 5=
BExample: if and
EvaluateA + B andA B.
1 2 2 3 3 0 3 5 3
0 ( 1) 1 2 4 5 1 3 9
+ + + + = = + + +
A B
1 2 2 3 3 0 1 1 3
0 ( 1) 1 2 4 5 1 1 1
= = A B
-
8/10/2019 1 Matrices and Determinants Sjk
11/40
Sums of matrices
. pera ons o ma r ces
Two matrices of the same order are said to be
or a on or su rac on.
Two matrices of different orders cannot be added or
su tracte , e.g.,
2 3 7 1 3 1
1 1 5 4 7 6
are con orma e or a t on or su tract on.
-
8/10/2019 1 Matrices and Determinants Sjk
12/40
Scalar multi lication
. pera ons o ma r ces
Let be any scalar andA = [aij] is an m n matrix. Then= ij or , , .e., eac e emen n
A is multiplied by .
0 1 4=
AExample: . Evaluate 3A.
3 3 0 3 1 3 4 0 3 12= = A
, , . ., ij .
negative ofA. Note:
A = 0 is a zero matrix
-
8/10/2019 1 Matrices and Determinants Sjk
13/40
. pera ons o ma r ces
MatricesA,B and C are conformable,
A + B = B + A
A + B +C = A +B +C
(commutative law)
associative law
(A +B) = A + B, where is a scalar
an you prove t em
-
8/10/2019 1 Matrices and Determinants Sjk
14/40
. pera ons o ma r ces
=
Let C=A +B, so cij = aij + bij.
.
Consider cij = (aij + bij ) = aij + bij, we have, C=
+
Since C = (A +B), so (A +B) = A + B
-
8/10/2019 1 Matrices and Determinants Sjk
15/40
Matrix multi lication
. pera ons o ma r ces
IfA = [aij] is a m p matrix andB = [bij] is ap nma r x, en s e ne as a ma r x = ,
where C= [cij] with
p
1 1 2 21
...=
ij ik kj i j i j ip pj
k
1 2 3 =A
1 2
2 3
=BExam le: , and C = AB.
, .
5 0 Evaluate c21.
1 2
2 30 1 4
5 0
21 0 ( 1) 1 2 4 5 22= + + =c
-
8/10/2019 1 Matrices and Determinants Sjk
16/40
Matrix multi lication
. pera ons o ma r ces
1 2 3 =
1 2 =
0 1 4 5 0
, , .
1 1 2 2 3 5 18c = + + =
12
21
1 21 2 2 3 3 0 81 2 3
2 30 ( 1) 1 2 4 5 220 1 4
c
c
= + + = = + + =
22 0 2 1 3 4 0 3c = + + =
1 2
2 30 1 4 22 3
5 0
C AB= = =
-
8/10/2019 1 Matrices and Determinants Sjk
17/40
Matrix multi lication
. pera ons o ma r ces
In particular,A is a 1 m matrix andB is a mma r x, .e.,
[ ]11 12 1...= mA a a a
11
21
= M
b
B
1 mb
then C = AB is a scalar.
1 1 11 11 12 21 1 1...= = + + +m
k k m mC a b a b a b a b
-
8/10/2019 1 Matrices and Determinants Sjk
18/40
. pera ons o ma r ces
BUTBA is a m m matrix!
11 11 11 11 12 11 1
21 21 11 21 12 21 1...
= =
K m
m
b b a b a b a
b b a b a b aBA a a a
1 1 11 1 12 1 1
m m m m mb b a b a b a
SoAB BA in general !
-
8/10/2019 1 Matrices and Determinants Sjk
19/40
Pro erties:
. pera ons o ma r ces
MatricesA,B and C are conformable,
A(B + C) = AB + AC
A + B C = AC+BC
A(BC) = (AB) C
AB BA in general
AB = 0 NOT necessaril im l A = 0 orB = 0
AB = ACNOT necessarily implyB = C
-
8/10/2019 1 Matrices and Determinants Sjk
20/40
-
8/10/2019 1 Matrices and Determinants Sjk
21/40
. ypes o matr ces
Identity matrix
The inverse of a matrix
The transpose of a matrix
ymmetr c matr x
Orthogonal matrix
-
8/10/2019 1 Matrices and Determinants Sjk
22/40
1.3 Types of matrices
= >
Identity matrix
,
called upper triangular, i.e., 11 12 1
22 20
K n
n
a a a
a a
0 0
M O
nna
A square matrix whose elements aij = 0, for i
-
8/10/2019 1 Matrices and Determinants Sjk
23/40
1.3 Types of matrices
=
Identity matrix
, . ., , , . .,
11 0 0
0 0
Ka
a
0 0
=
M O
nn
D
a
11 22diag[ , ,..., ]= nnD a a a
is called a diagonal matrix, simply
-
8/10/2019 1 Matrices and Determinants Sjk
24/40
1.3 Types of matrices
Identity matrix
In particular,a
11= a
22= = a
nn= 1
, the matrix iscalled identity matrix.
Properties:AI = IA = A
1 0 0 Examp es o i entity matrices: an
0 1 0 1 0
0 0 1
-
8/10/2019 1 Matrices and Determinants Sjk
25/40
1.3 Types of matrices
Special square matrix
. ,
andB such thatAB = BA, thenA andB are said to becommute.
Can you suggest two matrices that must commute with asquare ma r xAns:Aitself,theidentitymatrix,..
IfA andB such thatAB = -BA, thenA andB are said to
e ant -commute.
-
8/10/2019 1 Matrices and Determinants Sjk
26/40
1.3 Types of matrices
= =
e nverse o a matr x
,
called the inverse ofA (symbol:A-1); andA is called theinverse ofB s mbol:B-1 .
6 2 3
1 1 0B
=
1 2 3
1 3 3A
=Exam le:1 0 1
ShowB is the the inverse of matrixA.
1 2 4
1 0 0
0 1 0AB BA
= =Ans: Note that0 0 1 Can you show the
details?
-
8/10/2019 1 Matrices and Determinants Sjk
27/40
1.3 Types of matrices
e transpose o a matr x
columns of a matrixA is called the transpose ofA (writeAT).
Example:1 2 3
4 5 6
=
A
1 4 The transpose ofA is 2 5
3 6
=
TA
For a matrixA = [aij], its transposeAT= [bij], where bij= aji.
-
8/10/2019 1 Matrices and Determinants Sjk
28/40
1.3 Types of matrices
ymmetr c matr x
T= = , . .,
aij for all i andj.
mus e symme r c. y
1 2 3
2 4 5
= .3 5 6
T
-
- ,i.e., aji = -aij for all i andj.
- must e s ew-symmetric. W y?
-
8/10/2019 1 Matrices and Determinants Sjk
29/40
1.3 Types of matrices
Orthogonal matrix
T= T = T = , . .,
A-11/ 3 1/ 6 1/ 2
Example: prove that is orthogonal.1/ 3 2 / 6 0
1/ 3 1/ 6 1/ 2
=
A
Since, . Hence,AAT= ATA = I.1/ 3 1/ 3 1/ 3
1/ 6 2 / 6 1/ 6TA
=
1/ 2 0 1/ 2 Can you show thedetails?
Well see that orthogonal matrix represents a
rotation in fact!
-
8/10/2019 1 Matrices and Determinants Sjk
30/40
1.4 Properties o matrix
(AB
)
-1 = B-1A-1
AT T= A and A T= AT
(A + B)T= AT + BT
(AB)T= BTAT
-
8/10/2019 1 Matrices and Determinants Sjk
31/40
1.4 Properties o matrix
-1 = -1 -1 .
Since (AB) (B-1A-1) = A(B B-1)A-1 = Iand
(B-1A-1) (AB) = B-1(A-1A)B = I.
-1 -1 , .
-
8/10/2019 1 Matrices and Determinants Sjk
32/40
1.5 Determinants
Determinant of order 2
Consider a 2 2 matrix: 11 12a a
A =
21 22
, ,
evaluated by
11 1211 22 12 21
21 22
| | a a a a a aa a= =
-
8/10/2019 1 Matrices and Determinants Sjk
33/40
Determinant of order 2
1.5 Determinants
easy to remember (for order 2 only)..
11 12
11 22 12 21
21 22
| |a a
A a a a a
a a
= =+
1 2
-
3 4
1 21 4 2 3 2= =
-
8/10/2019 1 Matrices and Determinants Sjk
34/40
1.5 Determinants
order.
. every e ement o a row co umn s zero,
e.g., , then |A| = 0.
1 2
1 0 2 0 0= =
T
determinant of a matrix = thatof its trans ose.
3. |AB| = |A||B|
-
8/10/2019 1 Matrices and Determinants Sjk
35/40
1.5 Determinants
matrix is either +1 or1. T , .
Since|AAT| = |A||AT | = 1
and|AT| = |A|
, so|A|2 = 1
or|A| =
.
-
8/10/2019 1 Matrices and Determinants Sjk
36/40
1.5 Determinants
11 12a a
21 22a a
Its inverse can be written as 22 121 1a a
A
=
21 11a a
1 0 =1 2
The determinant of A is -2
Hence, the inverse of A is 1 1 01/ 2 1/ 2A =
How to in an inverse or a 3x3 matrix?
-
8/10/2019 1 Matrices and Determinants Sjk
37/40
1.5 Determinants of order 31 2 3
Consider an example: 4 5 6
7 8 9
A=
Its determinant can be obtained by:
4 5 1 2 1 24 5 6 3 6 9
7 8 7 8 4 57 8 9
A= = +
( ) ( ) ( )3 3 6 6 9 3 0= + =
ou are encourage o n e e erm nan y us ng o er
rows or columns
-
8/10/2019 1 Matrices and Determinants Sjk
38/40
1.6 Inverse of a 33 matrix1 2 3
Cofactor matrix of 0 4 5
1 0 6
A=
The cofactor for each element of matrix A:
4 5 0 5 0 4 110 6
121 6
131 0
2 3 1 3 1 221
0 6 22
1 623
1 0
2 3 1 3 1 231 4 5
= = 320 5
= = 33 0 4= =
-
8/10/2019 1 Matrices and Determinants Sjk
39/40
1.6 Inverse of a 33 matrix
Cofactor matrix of is then given by:0 4 51 0 6
A
=
12 3 2
2 5 4
-
8/10/2019 1 Matrices and Determinants Sjk
40/40
1.6 Inverse of a 33 matrix1 2 3
Inverse matrix of is given by:0 4 5
1 0 6
A=
1
24 5 4 24 12 21 1
12 3 2 5 3 5
T
A
= = 2 5 4 4 2 4
5 22 3 22 5 22
=