row echlon , reduced row echlon, hermitian, skew hermitian matrices etc. according to GTU SYLLABUS (VCLA)

1. SYSTEM OF LINEAR
EQUATIONS AND
MATRICES
Vector Calculus and
Linear Algebra

2. Group Members:
 140280117001 ADITYA VAISHAMPAYAN
 140280117002 NISHANT AHIR
 140280117004 BALDANIYA RAVI KUMAR
 140280117005 BHAVSAR MEET D
 140280117007 MARGESH DARJI
 140280117008 DARJI ROHAN RAKESH
 140280117009 DESAI HARSH DINESHBHAI
 140280117010 DHRUVAL NALIN SHAH
 140280117011 DODIYA VEDANG T
 140280117012 GHOGHARI NITIN H

3. Matrix :
 A rectangular array (arrangement) of mn numbers (real or
complex) in m rows and n columns is called a matrix of order m by
n written as m×n.
 A m×n matrix is usually written as
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
⋯
⋯
⋯
𝑎1𝑛
𝑎2𝑛
𝑎3𝑛
⋮ ⋮ ⋮ ⋮
𝑎 𝑚1 𝑎 𝑚2 𝑎 𝑚3 ⋯ 𝑎 𝑚𝑛 𝑚×𝑛
 This matrix is denoted in a simple form as
𝐴 = 𝑎𝑖𝑗 𝑚×𝑛
Where 𝑎𝑖𝑗 is the element in the 𝑖 𝑡ℎ
row and 𝑗 𝑡ℎ
column

4. Type Of Matrices :
1. Row Matrix :
A matrix which has only one row is called a row matrix or
row vector.
i.e.
1 −1 2 3 1×4 2 5 1×2
2. Column Matrix :
A matrix which has only one column is called a column
matrix or column vector.
i.e.
1
2
3 3×1
−1
1
0
4 4×1

5. 3. Square Matrix :
A matrix in which the number of rows is equal to
the number of columns is called a square matrix.
i.e.
1 −1 2
3 1 4
5 0 6 3×3
1 2
−1 4 2×2
4. Null or Zero Matrix :
A matrix in which each element is equal to zero
is called a null matrix and is denoted by O.
i.e.
0 0 0
0 0 0
0 0 0
0 0
0 0

6. 5. Diagonal Matrix :
A square matrix is called a diagonal matrix if all its
non diagonal elements are zero.
i.e.
1 0 0
0 2 0
0 0 4
3 0
0 5
6. Scalar Matrix :
A diagonal matrix whose all diagonal elements are
equal is called a scalar matrix.
i.e.
3 0 0
0 3 0
0 0 3
5 0
0 5

7. 7. Identity or Unit Matrix :
A diagonal matrix whose all diagonal elements are
unity (1) is called a unit or identity matrix and is
denoted by I.
i.e.
1 0 0
0 1 0
0 0 1
1 0
0 1
8. Upper Triangular Matrix :
A square matrix in which all the entries below the
diagonal are zero is called an upper triangular matrix.
i.e.
4 −3 2
0 3 5
0 0 1
1 3
0 2

8. 10. Trace of a Square Matrix :
The sum of all the diagonal elements of a square matrix is called the trace
of a matrix.
i.e.
If A =
−1 2
3 5
7 0
−8 4
1 2
4 −2
7 −3
1 0
then
Trace of 𝐴 = −1 + 5 + 7 + 0
= 11
 Note : If 𝐴 = 𝑎𝑖𝑗 𝑛×𝑛
then
Trace of 𝐴 = 𝑎11 + 𝑎22 + 𝑎33 + ------ +𝑎 𝑚𝑛
=
𝑖=1
𝑛
𝑎𝑖𝑖

9. 11. Transpose of a Matrix :
The matrix obtained by interchanging the rows and columns
of a given matrix A is called the transpose of A and is denoted
by 𝐴 𝑇 or 𝐴′.
i.e.
𝐼𝐹 𝐴 =
1 2 3
4 5 6 2×3
𝑡ℎ𝑒𝑛 𝐴 𝑇 =
1 4
2 5
3 6 3×2
 Properties :
I. 𝐴 𝑇 𝑇
= 𝐴
II. 𝐴𝐵 𝑇
= 𝐵 𝑇
𝐴 𝑇
III. 𝐴 + 𝐵 𝑇 = 𝐴 𝑇 + 𝐵 𝑇
IV. 𝐾𝐴 𝑇
= 𝐾𝐴 𝑇
where K is any scalar.

10. 12. Determinant of a Matrix :
If A is a square matrix then determinant of A is represented as 𝐴 or
det(A).
i.e.
𝐼𝐹 𝐴 =
2 1 −3
4 3 5
5 4 9
𝑡ℎ𝑒𝑛 det 𝐴 =
2 1 −3
4 3 5
5 4 9
13. Singular and Non-singular Matrix :
A square matrix A is called singular if det(A)=0 and non-singular or
invertible if det(A)≠0.
i.e.
𝐿𝑒𝑡 𝐴 =
2 1 −3
4 3 5
5 4 9
𝑡ℎ𝑒𝑛 det 𝐴 =
2 1 −3
4 3 5
5 4 9
⟹ 𝐴 = 2 27 − 20 − 1 36 − 25 − 3(16 − 15)
= 14 − 11 − 3 = 0
Hence, A is a singular matrix.

11. Elementary Transformations :
 Elementary Row Transformations :
1) 𝑅𝑖𝑗 ∶ Interchange of 𝑖 𝑡ℎ and 𝑗 𝑡ℎ row.
2) 𝐾𝑅𝑖 ∶ Multiplication of 𝑖 𝑡ℎ rowby K.
3) 𝑅𝑗 + 𝐾𝑅𝑖 ∶ Addition of K times the 𝑖 𝑡ℎ
row to the 𝑗 𝑡ℎ
row.
 Elementary column Transformations :
1) 𝐶𝑖𝑗 ∶ Interchange of 𝑖 𝑡ℎ and 𝑗 𝑡ℎ column.
2) 𝐾𝐶𝑖 ∶ Multiplication of 𝑖 𝑡ℎ column by K.
3) 𝐶𝑗 + 𝐾𝐶𝑖 ∶ Addition of K times the 𝑖 𝑡ℎ
column to the
𝑗 𝑡ℎ
column.

12.  Equivalent Matrices :
Two matrices A and B of same order are said to be
equivalent matrices if one of them is obtained from the
other by elementary transformation.
Symbolically, we can write A ~ B.
 Row Echelon form :
A given matrix is said to be in row echelon form if it
satisfies the following properties:
I. The zero rows of the matrix if they exists occur below
the non-zero rows of the matrix.
II. The first non-zero element in any non-zero row of the
matrix must be equal to unity (1). We call this
a leading 1.

13. III. In any two successive rows that do not consists
entirely of zeros, the leading 1 in the lower row
occurs farther to the right than the leading 1 in the
higher row.
 The following matrices are in row echelon form :
1 1 0
0 1 0
0 0 0
,
1 4 −3 7
0 1 6 2
0 0 1 5
,
0 1 2 6 0
0 0 1 −1 0
0 0 0 0 1

14.  Reduced Row Echelon Form :
A given matric is said to be in reduced row echelon
form if it satisfies the following properties :
I. A matrix is of necessity in row echelon form.
II. Each column that contains a leading 1 has zero
everywhere else in that column.
 The following matrices are in reduced row echelon
form.
0 0
0 0
1 0 0
0 1 0
0 0 1
1 0 0 4
0 1 0 7
0 0 1 −1
0 1
0 0
−2 0 1
0 1 3
0 0
0 0
0 0 0
0 0 0

15. Notes:
(1) Every matrix has an unique reduced row echelon form.
(2) A row-echelon form of a given matrix is not unique.
(Different sequences of row operations can produce
different row-echelon forms.)

16. System of Non-Homogeneous Linear Equations :
A system of m linear equations in n unknowns can be
written as
𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ ⋯ ⋯ + 𝑎1𝑛 𝑥 𝑛 = 𝑏1
𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ ⋯ ⋯ + 𝑎2𝑛 𝑥 𝑛 = 𝑏2
⋮ ⋮ ⋮ = ⋮
𝑎 𝑚1 𝑥1 + 𝑎 𝑚2 𝑥2 + ⋯ ⋯ ⋯ + 𝑎 𝑚𝑛 𝑥 𝑛 = 𝑏 𝑚
The above system can be written in a matrix form as
𝑎11 𝑎12
𝑎21 𝑎22
⋯ 𝑎1𝑛
⋯ 𝑎2𝑛
⋮ ⋮
𝑎 𝑚1 𝑎 𝑚2
⋮
⋯ 𝑎 𝑚𝑛
𝑥1
𝑥2
⋮
𝑥 𝑛
=
𝑏1
𝑏2
⋮
𝑏 𝑚
Or Simply A X = B

17. Where, A =
𝑎11 𝑎12
𝑎21 𝑎22
⋯ 𝑎1𝑛
⋯ 𝑎2𝑛
⋮ ⋮
𝑎 𝑚1 𝑎 𝑚2
⋮
⋯ 𝑎 𝑚𝑛
is called coefficient matrix of order m×n.
𝑋 =
𝑥1
𝑥2
⋮
𝑥 𝑛
𝑎𝑛𝑑 𝐵 =
𝑏1
𝑏2
⋮
𝑏 𝑚
is any vector of order n × 1 & 𝑚 × 1
respectively.
𝐴 : 𝐵 =
𝑎11 𝑎12
𝑎21 𝑎22
⋯ 𝑎1𝑛 : 𝑏1
⋯ 𝑎2𝑛 : 𝑏2
⋮ ⋮
𝑎 𝑚1 𝑎 𝑚2
⋮ ∶ ⋮
⋯ 𝑎 𝑚𝑛 : 𝑏 𝑚
is the augmented matrix of the given system of linear equations.

18. Solution of system of Linear
Equations :
For a system of m linear equations in n unknowns, there are three
possibilities of the solutions to the system
I. The system has unique solution.
II. The system has infinite solutions.
III. The solution has no solution.
 When the system of linear equations has one or more solutions,
the system is said to be consistent otherwise it is inconsistent.

19. 1. Gauss-Jordan Elimination :
Reducing the augmented matrix to reduced row
echelon form is called Gauss-Jordan Elimination.
2. Gauss-Elimination :
Reducing the augmented matrix to “row
echelon form” and then stopping is called Gaussian
Elimination.

20. (4) For a square matrix, the entries a11, a22, …, ann are called
the main diagonal entries.
1.2 Gaussian Elimination and Gauss-Jordan
Elimination
 mn matrix:












mnmmm
n
n
n
aaaa
aaaa
aaaa
aaaa





321
3333231
2232221
1131211
rowsm
columnsn
(3) If , then the matrix is called square of order n.nm 
 Notes:
(1) Every entry aij in a matrix is a number.
(2) A matrix with m rows and n columns is said to be of size mn .

21.  a system of m equations in n variables:
mnmnmmm
nn
nn
nn
bxaxaxaxa
bxaxaxaxa
bxaxaxaxa
bxaxaxaxa









332211
33333232131
22323222121
11313212111













mnmmm
n
n
n
aaaa
aaaa
aaaa
aaaa
A





321
3333231
2232221
1131211











mb
b
b
b

2
1











nx
x
x
x

2
1
bAx Matrix form:

22. Augmented matrix:
][3
2
1
321
3333231
2232221
1131211
bA
b
b
b
b
aaaa
aaaa
aaaa
aaaa
mmnmmm
n
n
n


















A
aaaa
aaaa
aaaa
aaaa
mnmmm
n
n
n


















321
3333231
2232221
1131211
Coefficient matrix:

23.  Elementary row operation:
jiij RRr :(1) Interchange two rows.
ii
k
i RRkr )(:)((2) Multiply a row by a nonzero constant.
jji
k
ij RRRkr )(:)((3) Add a multiple of a row to another row.
 Row equivalent:
Two matrices are said to be row equivalent if one can be obtained
from the other by a finite sequence of elementary row operation.

24.  Ex 2: (Elementary row operation)










1432
4310
3021










1432
3021
4310 12r











2125
0331
1321











2125
0331
2642 )(
1
2
1
r











81330
1230
3421











2512
1230
3421 )2(
13

r

25.  Ex 3: Using elementary row operations to solve a system
17552
53
932



zyx
zy
zyx
17552
43
932



zyx
yx
zyx
Linear System













17552
4031
9321












17552
5310
9321
Associated
Augemented Matrix
Elementary
Row Operation












1110
5310
9321
221
)1(
12 )1(: RRRr 
331
)2(
13 )2(: RRRr 
1
53
932



zy
zy
zyx

26. 332
)1(
23 )1(: RRRr 
Linear System









 
4200
5310
9321
2
1
1



z
y
x









 
2100
5310
9321
Associated
Augemented Matrix
Elementary
Row Operation
42
53
932



z
zy
zyx
33
)
2
1
(
3 )
2
1
(: RRr 
2
53
932



z
zy
zyx

27. (1) All row consisting entirely of zeros occur at the bottom
of the matrix.
(2) For each row that does not consist entirely of zeros,
the first nonzero entry is 1 (called a leading 1).
(3) For two successive (nonzero) rows, the leading 1 in the higher
row is farther to the left than the leading 1 in the lower row.
 Reduced row-echelon form:
(4) Every column that has a leading 1 has zeros in every position
above and below its leading 1.

28. form)echelon
-row(reduced
form)
echelon-(row
 Ex 4: (Row-echelon form or reduced row-echelon form)












10000
41000
23100
31251








0000
3100
5010









 
0000
3100
2010
1001











3100
1120
4321










4210
0000
2121
form)
echelon-(row
form)echelon
-row(reduced










2100
3010
4121

29. Solutions of a system of linear Equations :
There are only two possibilities for the solution of homogenous
linear system.
I. The system has exactly one solution.
i.e. 𝑥1 = 0, 𝑥2 = 0, ⋯ ⋯ ⋯ , 𝑥 𝑛 = 0
This solution is called the trivial solution.
II. The system has infinite solutions, this solution is called the
non-trivial solution.
 Note : The system of equation has non-trivial solution if det(A) =
0.

30. Cramer’s rule :
 Theorem-1 :
If A X = B is a system of n linear equations in n unknowns such
that det(A) ≠ 0, then the system has a unique solution. This
solution is
𝑥1 =
det(𝐴1)
det(𝐴)
, 𝑥2 =
det(𝐴2)
det(𝐴)
, ⋯ ⋯ ⋯ , 𝑥 𝑛 =
det(𝐴 𝑛)
det(𝐴)
where 𝐴𝑗 is the matrix obtained by replacing the entries in the
𝑗 𝑡ℎ
column of A by the entries in the matrix. B =
𝑏1
𝑏2
⋮
𝑏 𝑛
 Note : Cramer’s rule can’t be used for a system AX=B in which
det(A) = 0.

31. Determinant of an Upper Triangular
Matrix :
Let A =
𝑎11 𝑎12
0 𝑎22
𝑎13 𝑎14
𝑎23 𝑎24
0 0
0 0
𝑎33 𝑎34
0 𝑎44
det (A) =
𝑎11 𝑎12
0 𝑎22
𝑎13 𝑎14
𝑎23 𝑎24
0 0
0 0
𝑎33 𝑎34
0 𝑎44
= 𝑎11
𝑎22 𝑎23 𝑎24
0 𝑎33 𝑎34
0 0 𝑎44
= 𝑎11 𝑎22
𝑎33 𝑎34
0 𝑎44
= 𝑎11 ∗ 𝑎22 ∗ 𝑎33 ∗ 𝑎44

32.  Theorem-2 :
If A is an 𝑛 × 𝑛 triangular matrix ( upper triangular, lower
triangular or diagonal ) then det(A) is the product of the entries
on the main diagonal of the matrix. That is
det (A) = 𝑎11 ∗ 𝑎22 ∗ 𝑎33 ⋯ ⋯ ⋯ 𝑎 𝑚𝑛
i.e.
If A =
2 7
0 −3
−3 8
7 5
0 0
0 0
6 7
0 9
then
det(A) = (2) (−3) (6) (9)
= −324

33. Minors and Cofactors :
1. Minor of an element of a Determinant :
If det A =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
, 𝑡ℎ𝑒𝑛
Minor of 𝑎11 =
𝑎22 𝑎23
𝑎32 𝑎33
Minor of 𝑎22 =
𝑎11 𝑎13
𝑎31 𝑎33
Minor of 𝑎32 =
𝑎11 𝑎13
𝑎21 𝑎23

34. 2. Cofactor of an element of a Determinant :
If det A =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
, 𝑡ℎ𝑒𝑛
Cofactor of 𝑎11 = −1 1+1
𝑎22 𝑎23
𝑎32 𝑎33
Cofactor of 𝑎22 = −1 1+1
𝑎11 𝑎13
𝑎31 𝑎33
Cofactor of 𝑎32 = −1 1+1
𝑎11 𝑎13
𝑎21 𝑎23

35. Adjoint of square Matrix :
 The transpose of the matrix of the cofactors is called the adjoint of the
matrix.
𝐿𝑒𝑡 𝐴 =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
The matrix formed by the cofactors of the element of A is
𝐶𝐴 =
𝐴11 𝐴12 𝐴13
𝐴21 𝐴22 𝐴23
𝐴31 𝐴32 𝐴33
where 𝐴𝑖𝑗 is the cofactor of 𝑎𝑖𝑗.
Then, adj A = 𝐶𝐴
𝑇 =
𝐴11 𝐴21 𝐴31
𝐴12 𝐴22 𝐴32
𝐴13 𝐴23 𝐴33

36. Invertible Matrix :
If A is a square matrix and if a matrix B of the same order can
be found such that
AB = BA = I
then A is said to be invertible and B is called an inverse of A.
i.e. 𝐵 = 𝐴−1.

37. Inverse of a Matrix by Determinant Method :
If A is non-singular square matrix, then
𝐴−1
=
1
det(𝐴)
𝑎𝑑𝑗 𝐴
Note :
(1) If det (A) = 0 then A is not invertible.
i.e. 𝐴−1 does not exist.
(2) If A =
𝑎 𝑏
𝑐 𝑑
𝑡ℎ𝑒𝑛 𝑎𝑑𝑗 𝐴 =
𝑑 −𝑏
−𝑐 𝑎
∴ 𝐴−1 =
1
𝑎𝑑−𝑏𝑐
𝑑 −𝑏
−𝑐 𝑎

38. Inverse of a matrix by Elementary
Transformation :
Gauss-Jordan Method :
Let A be a given non-singular matrix of order n.
Let 𝐼 𝑛 be the unit matrix of order n.
To find 𝐴−1, take the matrix 𝐴 ∶ 𝐼 𝑛 and reduce it with
the help of a series of row operations to the form
𝐼 𝑛 ∶ 𝐵 . Then 𝐵 = 𝐴−1.

39. Inverses and Powers of Diagonal Matrices :
A general n× 𝑛 diagonal matrix D can be written as
D =
𝑑1 0
0 𝑑2
⋯ 0
⋯ 0
⋮ ⋮
0 0
⋮
⋯ 𝑑 𝑛
A diagonal matrix is invertible if and only if all of its diagonal
entries are non-zero.
In this case D is
𝐷−1
=
1 𝑑1 0
0 1 𝑑2
⋯ 0
⋯ 0
⋮ ⋮
0 0
⋮
0 1 𝑑 𝑛

40. 𝐷 𝐾 =
𝑑1
𝐾
0
0 𝑑2
𝐾
⋯ 0
⋯ 0
⋮ ⋮
0 0
⋮
0 𝑑 𝑛
𝐾
i.e.
IF A =
1 0 0
0 −3 0
0 0 2
, then
𝐴−1 =
1 0 0
0 −1
3 0
0 0 1
2
𝐴5 =
1 0 0
0 −243 0
0 0 32

41. Rank of Matrix :
A matrix is said to be of rank r if it satisfies the following
properties :
I. There is at least one minor of order r which is not zero
II. Every minor of order (r+1) is zero.
 Rank of matrix A is denoted by 𝜌(𝐴).
 There are three methods for finding the rank of a matrix
1. Rank of a matrix by Determinant method.
2. Rank of a matrix by row Echelon form.
3. Rank of a matrix by Normal form.

42.  Note :
I. If A is zero matrix then 𝜌 𝐴 = 0.
II. If A is not a zero matrix then 𝜌 𝐴 ≥ 1.
III. If A is a non-singular n× 𝑛 matrix then 𝜌 𝐴 = 𝑛.
IV. 𝜌 𝐼 𝑛 = 𝑛 , where 𝐼 𝑛 is 𝑛 × 𝑛 unit matrix .
V. If A is 𝑚 × 𝑛 matrix then 𝜌 𝐴 ≤ 𝑚𝑖𝑛 𝑚 , 𝑛 .

43. Rank of Matrix by Row Echelon Form :
The rank of a matrix in row echelon form is equal to the
number of non-zero rows of the matrix.
i.e.
Rank of matrix = Number of non-zero rows
e.g.
𝐼𝑓 𝐴 =
1 2
0 0
5 0
1 0
0 0 0 0
𝑡ℎ𝑒𝑛
𝜌 𝐴 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑜𝑛 − 𝑧𝑒𝑟𝑜 𝑟𝑜𝑤𝑠
𝜌 𝐴 = 2

44. Rank of Matrix by Normal Form :
If a matrix A is of the form
𝐼𝑟 0
0 0
𝑜𝑟
𝐼𝑟
0
𝑜𝑟 𝐼𝑟 0 𝑜𝑟 𝐼𝑟
Then it called normal form of A.
Note : Rank of A = r.
SOME SPECIAL MATRIX :
1. Symmetric Matrix :
A Square matrix A= 𝑎𝑖𝑗 𝑛×𝑛
is called symmetric matrix
if 𝑎𝑖𝑗 = 𝑎𝑗𝑖 ∀ 𝑖 & 𝑗. OR A = 𝐴 𝑇
i.e.
7 −3
−3 5
1 4 5
4 −3 0
5 0 7

45. 2. Skew Symmetric Matrix :
A Square matrix A= 𝑎𝑖𝑗 𝑛×𝑛
is called Skew symmetric matrix if
𝑎𝑖𝑗 = −𝑎𝑗𝑖 ∀ 𝑖 & 𝑗. OR A = −𝐴 𝑇
If 𝑖 = 𝑗 , 𝑎𝑖𝑖 = −𝑎𝑖𝑖
2𝑎𝑖𝑖 = 0
𝑎𝑖𝑖 = 0 ∀ 𝑖
Thus the diagonal element of a skew symmetric matrix are all zero.
i.e.
0 1 4
−1 0 −3
−4 3 0
0 ℎ 𝑔
−ℎ 0 𝑓
−𝑔 −𝑓 0
3. Orthogonal Matrix :
A square matrix A is called orthogonal if 𝐴 𝑇
= 𝐴−1
𝑜𝑟 𝐴𝐴 𝑇 = 𝐴 𝑇 𝐴 = 𝐼.

46. Complex Matrix :
If all the element of a matrix are real numbers then it is called a
real matrix.
If at least one element of a matrix is a complex number 𝑎 + 𝑖𝑏
where 𝑎 , 𝑏 are real numbers and 𝑖 = −1 then the matrix is
called a complex matrix.
i.e.
2 + 𝑖 0 −5
3 −𝑖 4 + 2𝑖

47.  Conjugate of a matrix :
The matrix obtained from any given matrix A by replacing its
element by the corresponding complex conjugates is called the
conjugate of A and is denoted by 𝐴.
i.e.
𝐴 =
2 + 3𝑖 −4𝑖
5 7 − 2𝑖
⇒ 𝐴 =
2 − 3𝑖 4𝑖
5 7 + 2𝑖
 Transposed conjugate of a matrix :
The transpose of the conjugate matrix of a matrix A is called
the transposed conjugate of A and is denoted by 𝐴∗
i.e.
𝐴∗ = 𝐴
𝑇
= 𝐴 𝑇

48. Hermitian Matrix :
A square matrix 𝐴 = 𝑎𝑖𝑗 is said to be Hermitian if 𝑎𝑖𝑗 = 𝑎𝑖𝑗 ∀ 𝑖 & 𝑗
Or
𝐴 = 𝐴∗ where 𝐴∗ = 𝐴
𝑇
i.e.
1 2 − 3𝑖 4 + 𝑖
2 + 3𝑖 3 1 + 2𝑖
4 − 𝑖 1 − 2𝑖 7
Note : In a Hermitian matrix, the diagonal elements are real

49. Skew Hermitian Matrix :
A square matrix 𝐴 = 𝑎𝑖𝑗 is said to be skew Hermitian matrix if
𝑎𝑖𝑗 = −𝑎𝑖𝑗 ∀ 𝑖 & 𝑗
Or
𝐴 = 𝐴∗ where 𝐴∗ = 𝐴
𝑇
i.e.
2𝑖 3 + 𝑖 5
−3 + 𝑖 0 −1 − 𝑖
−5 1 − 𝑖 −𝑖
Note : In a skew Hermitian matrix , the diagonal elements are zero or
purely imaginary numbers.

50.  Unitary Matrix : A square matrix A is said to
be unitary if
𝐴∗ = 𝐴−1 or 𝐴∗ 𝐴 = 𝐴𝐴∗ = 𝐼
Where 𝐴∗
= 𝐴
𝑇

51. THANK YOU