Lecture 01.pdf Linear algebra linear algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces, linear transformations, and matrices. It’s the language of systems—whether you're solving equations, modeling physical phenomena, or analyzing data.

1. Linear Algebra
BSCS (Semester Ⅱ)
Lecture# 01
May 29, 2020
Section -B
Amna Tahir
GCU, Lahore

2. Topics Covered
▪Matrices (Definition, Types)
▪Algebra of matrices (Addition, Subtraction,
Transpose, Scalar multiplication)
▪Matrix multiplication (Power of matrices, Periodic,
Involutory, Idempotent & Nilpotent matrices)
▪Determinant, Adjoint & Inverse (2×2)

3. Today’s Agenda
▪Determinants of order 3
▪Evaluation technique
▪FindingInverse

4. Determinant:
A unique value associated with a square matrix
of order n.
❖Determinant of order 1:
A = [a]
??????=�=�
❖Determinant of order 2:
A =
��
��
??????=
��
��
= ad−bc

5. DETERMINANT OF ORDER 3:
The given determinant is expanded by 1
st
column.
Note:
•A determinant mayexpand by any row or column.
•To expand the determinant, choose a row/column with maximum
number of zeros.

6. Expansion Technique
(Row Expansion)
Example:A =
130
264
−102
•Expansion by 1
st
row:
??????=
130
264
−102
= 1
64
02
−3
24
−12
+0
26
−10
= 1(12 –0 ) –3 (4 + 4) + 0(0 + 6)
=12–24+0= –12
•Expansionby2
nd
row:
??????=
130
264
−102
= –2
30
02
+6
10
−12
−4
13
−10
=−��
+−+
−+−
+−+

7. Expansion Technique
(Column Expansion)
Example:A =
130
264
−102
•Expansion by 1
st
column:
??????=
130
264
−102
= 1
64
02
−2
30
02
+(−1)
30
64
= 1(12 –0 ) –2 (6–0) –1 (12 –0 )
= 12 –12–12= –12
•Expansion by 2
nd
column:
??????=
130
264
−102
= –3
24
−12
+6
10
−12
−0
10
24
=−��
+−+
−+−
+−+

8. Questions
Note:Evaluate the following determinants.
1.
70.40.30.8
00.52.6
0 0−1.9
2.
0��
−�0�
−�−�0
3.
03−1
−30−4
140
4.
���
���
���

9. Finding Inverse

10. How To Find Inverse(??????
−1
)
•??????
−1
=
1
??????
adj A
•Usingelementary operations

11. Steps To Find Inverse(order 3)
•Step-1:??????≠0??????.�.??????
−1
exists.
•Step-2:Find cofactors ??????
��
•Step-3:Adj A = (MatrixofcofactorsofA)
T
•Step-4:??????
−1
=
1
??????
adj A

12. Cofactors
Example:A =
123
456
789
??????
11=(−1)
1+156
89
= –3
??????
12=(−1)
1+246
79
= –3
??????
13=(−1)
1+345
78
= –3
Similarly,
??????
21= 6 , ??????
22= −12,??????
23= 6
??????
31= –3 , ??????
32= 6, ??????
33= –3
•Cofactor of an
element ??????
��= ??????
��
•Everyelementofa
matrix hasitsown
cofactor.

13. Adjoint of A
Adj A =
??????
11??????
12??????
13
??????
21??????
22??????
23
??????
31??????
32??????
33
??????
AdjA=
−3−3−3
6−126
−36−3
??????
=
−36−3
−3−126
−36−3

14. Questions
Find the inverse(if exists) by using formula:
•A=
−112
3−11
−134
•A =
125
0−12
2410