Chapter 1 &
:MATRIX SYSTEM OF LINEAR
EQUATION
1.1 MATRIX Exp!
(a I
..B ( ). :(2,3]
A
= 36 =
Matrix · is a rectangular array of numbers.
·
the number in array are called entries
M is number of row
->
i) size of matrix(mxn)
n is number of column
ii) column matrix (column vector) -> B
iii) Row (row rector)
matrix -> c
iv) Entries of a matrix -> A [dij]
=
v)
square matrix -> (the
A diagonal is 3, 10)
Matrix Operations
2,).B
I ( Y,5]
Equal Matrix A
c exp:
=
:
·
if both matrixsame size =
the entries of a matrixsame
A B
-
i.
=
A:/2i7B) I
2
Addition and subtraction exp:
-
size of matrixmust same I
( i) ( ) (ii)
a B
+ =
-
-
=
(io)".o.)
3 Scalar Multiple of a Matrix Aexp.
is
<[A] [ cA] =
4 MatrixMultiplication -:
(5)()
I 1
B -1)
if A =
St),
·
pB=/ I
3x2
, 3 Partitioned Matrices Exp:
I I -Bit
a matrix can be subdivided or
partitioned into B 414
=
3
smaller matrices or submatrices: 0 -13 I
275 2
6 Transpose Matrix (AY exp!
I I I e
I
Anxm
2
SS
-
if Aman so the
transpose is A : A =
a 4
S 5 10 5
Trace
7 of a Matrix(tr)
-
if Ai s non matrix, then tr(A)= A,+dazt.... dun
expi
I -ii. I A(tr(A)
I 23
A 4
trace 6
=
1 + 10 +
21
+
-
= =
3
5
Rule of MatrixOperations & Inverse 6
Matrixoperation
a) Properties of
f)
Law of Exponents
not need to
AB and BA equal AB FBA if A is
square matrix, and s are integer, then
A "AS:Arts and (AV) Ars
=
invertible
b) Properties of MatrixArithmetic
(boleh terbalik)
B + A 9) Law of Exponents expiAB BA:In
A B
=
+
=
A(B c) + AB
=
+
AC if A
is invertible, then:
a(BC) =
(aB) c (ac)
= B ⑧ A "is
invertible and
(A)* A =
nonnegative
-X boleh-re
· A =LA")" for nonnegative n
-
hesti the 10 =
c)
Properties
"
Zero ·
if nonzero scalark, CKA)"=iA
A O:
+ O + A
= A
A A n)
0
Properties oft he transpose
n
=
A A
A - ⑧
(A) =
OA = O
LA B) +
*
= A +BY and (A-B):AT - BT
if cA 0
= then c 0
=
or A 0 (KAS KAY, =
K is a scalar
LABST BYAY
=
Multiplication Matrix (In=1nxn)
of Identity
d) invertibility
Invertibility Keterbalikkan
C
IfAi s man, a n dA In:A i) of transpose
I
then Alm:A a
transpose
If invertible matrix, then
A
e) Power Matrix Al is also can invertible
* -
matrix, (A 1)
-
If A
is square then (AT) =
A0 1
=
i) Invertible Matrix
Al A.A.A.... A (n times) is matrix and ifB issame size and
if A square
=
a
non singular
A=(A ) =A .A: A"(n times) square matrix whose AB:B A=1 i t said
then A invertible and B is inverse
determinant no
equal
to zero of A. Ifno such matrix B found, then A
is a
(invertible matrixcalled
singular. non
singular)
:
:MATRIX SYSTEM OF LINEAR
EQUATION
1.1 MATRIX Exp!
(a I
..B ( ). :(2,3]
A
= 36 =
Matrix · is a rectangular array of numbers.
·
the number in array are called entries
M is number of row
->
i) size of matrix(mxn)
n is number of column
ii) column matrix (column vector) -> B
iii) Row (row rector)
matrix -> c
iv) Entries of a matrix -> A [dij]
=
v)
square matrix -> (the
A diagonal is 3, 10)
Matrix Operations
2,).B
I ( Y,5]
Equal Matrix A
c exp:
=
:
·
if both matrixsame size =
the entries of a matrixsame
A B
-
i.
=
A:/2i7B) I
2
Addition and subtraction exp:
-
size of matrixmust same I
( i) ( ) (ii)
a B
+ =
-
-
=
(io)".o.)
3 Scalar Multiple of a Matrix Aexp.
is
<[A] [ cA] =
4 MatrixMultiplication -:
(5)()
I 1
B -1)
if A =
St),
·
pB=/ I
3x2
, 3 Partitioned Matrices Exp:
I I -Bit
a matrix can be subdivided or
partitioned into B 414
=
3
smaller matrices or submatrices: 0 -13 I
275 2
6 Transpose Matrix (AY exp!
I I I e
I
Anxm
2
SS
-
if Aman so the
transpose is A : A =
a 4
S 5 10 5
Trace
7 of a Matrix(tr)
-
if Ai s non matrix, then tr(A)= A,+dazt.... dun
expi
I -ii. I A(tr(A)
I 23
A 4
trace 6
=
1 + 10 +
21
+
-
= =
3
5
Rule of MatrixOperations & Inverse 6
Matrixoperation
a) Properties of
f)
Law of Exponents
not need to
AB and BA equal AB FBA if A is
square matrix, and s are integer, then
A "AS:Arts and (AV) Ars
=
invertible
b) Properties of MatrixArithmetic
(boleh terbalik)
B + A 9) Law of Exponents expiAB BA:In
A B
=
+
=
A(B c) + AB
=
+
AC if A
is invertible, then:
a(BC) =
(aB) c (ac)
= B ⑧ A "is
invertible and
(A)* A =
nonnegative
-X boleh-re
· A =LA")" for nonnegative n
-
hesti the 10 =
c)
Properties
"
Zero ·
if nonzero scalark, CKA)"=iA
A O:
+ O + A
= A
A A n)
0
Properties oft he transpose
n
=
A A
A - ⑧
(A) =
OA = O
LA B) +
*
= A +BY and (A-B):AT - BT
if cA 0
= then c 0
=
or A 0 (KAS KAY, =
K is a scalar
LABST BYAY
=
Multiplication Matrix (In=1nxn)
of Identity
d) invertibility
Invertibility Keterbalikkan
C
IfAi s man, a n dA In:A i) of transpose
I
then Alm:A a
transpose
If invertible matrix, then
A
e) Power Matrix Al is also can invertible
* -
matrix, (A 1)
-
If A
is square then (AT) =
A0 1
=
i) Invertible Matrix
Al A.A.A.... A (n times) is matrix and ifB issame size and
if A square
=
a
non singular
A=(A ) =A .A: A"(n times) square matrix whose AB:B A=1 i t said
then A invertible and B is inverse
determinant no
equal
to zero of A. Ifno such matrix B found, then A
is a
(invertible matrixcalled
singular. non
singular)
:
