Summary

Summary Matrix and linear equations

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Course
MAT523 (MAT523)

Institution
University Teknologi Mara

Matrices and linear equations are fundamental concepts in mathematics in field of linear algebra. Matrix is a rectangular array of numbers, symbols arranged in rows and columns. Matrices can be added, subtracted, and multiplied under certain conditions. Linear equation is an equation in which the...

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Uploaded on August 16, 2023
Number of pages 13
Written in 2023/2024
Type Summary

matrices
matrix addition
matrix multiple
matrix subtract

Institution
University Teknologi Mara
Course
MAT523 (MAT523)

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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)
:

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