Important Formulae: Matrices

 Page 1


1 Matrices
A matrix is a rectangular arra y of n um b ers, sym b ols, or expressions arranged in ro ws and
columns. If A = [a
ij
]
m×n
, then A is a matrix of order m×n , where:
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
Here, m is the n um b er of ro ws, n is the n um b er of columns, and a
ij
represen ts the elemen t
in the i -th ro w and j -th column.
2 Notation and Sp ecial Characters
Matrices often use sp ecial c haracters and notations to represen t their elemen ts and op er-
ations. Belo w is an explanation of the k ey sym b ols used in this do cumen t:
• a
ij
: The elemen t in thei -th ro w and j -th column of a matrix. The subscriptij indicates
the p osition, where i is the ro w index and j is the c olumn index.
• A
T
: The transp ose of matrix A . The sup erscript T denotes the op eration of in ter-
c hanging ro ws and columns.
• A
*
: The conjugate transp ose (or adjoin t) of matrix A . The sup erscript * indicates
that the matrix is first conjugated (complex conjugate of eac h elemen t) and then
transp osed.
• A : The conjugate of matrix A . The o v erline sym b ol denotes the op eration of taking
the complex conjugate of eac h elemen t.
• ? : The summation sym b ol, used in matrix m ultiplication and trace. F or example,
?
n
k=1
a
ik
b
kj
represen ts the sum of pro ducts for computing an elemen t in matrix m ulti-
plication.
• det(A) : The determinan t of matrix A . The notation det ? is a function that computes
a scalar v alue for a square matrix.
• |A| : An alternativ e notation for the determinan t of A , often used in the con text of
in v erses.
•
.
.
. : A sym b ol used in matrix notation to indicate that the pattern of elemen ts con-
tin ues diagonally (e.g., in a general m×n matrix).
3 Basic Op erations on Matrices
3.1 A ddition and Subtraction
Matrices can b e added or subtracted if they ha v e the same order. The follo wing prop erties
hold:
• Comm utativ e: A+B = B+ A
2
Page 2


1 Matrices
A matrix is a rectangular arra y of n um b ers, sym b ols, or expressions arranged in ro ws and
columns. If A = [a
ij
]
m×n
, then A is a matrix of order m×n , where:
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
Here, m is the n um b er of ro ws, n is the n um b er of columns, and a
ij
represen ts the elemen t
in the i -th ro w and j -th column.
2 Notation and Sp ecial Characters
Matrices often use sp ecial c haracters and notations to represen t their elemen ts and op er-
ations. Belo w is an explanation of the k ey sym b ols used in this do cumen t:
• a
ij
: The elemen t in thei -th ro w and j -th column of a matrix. The subscriptij indicates
the p osition, where i is the ro w index and j is the c olumn index.
• A
T
: The transp ose of matrix A . The sup erscript T denotes the op eration of in ter-
c hanging ro ws and columns.
• A
*
: The conjugate transp ose (or adjoin t) of matrix A . The sup erscript * indicates
that the matrix is first conjugated (complex conjugate of eac h elemen t) and then
transp osed.
• A : The conjugate of matrix A . The o v erline sym b ol denotes the op eration of taking
the complex conjugate of eac h elemen t.
• ? : The summation sym b ol, used in matrix m ultiplication and trace. F or example,
?
n
k=1
a
ik
b
kj
represen ts the sum of pro ducts for computing an elemen t in matrix m ulti-
plication.
• det(A) : The determinan t of matrix A . The notation det ? is a function that computes
a scalar v alue for a square matrix.
• |A| : An alternativ e notation for the determinan t of A , often used in the con text of
in v erses.
•
.
.
. : A sym b ol used in matrix notation to indicate that the pattern of elemen ts con-
tin ues diagonally (e.g., in a general m×n matrix).
3 Basic Op erations on Matrices
3.1 A ddition and Subtraction
Matrices can b e added or subtracted if they ha v e the same order. The follo wing prop erties
hold:
• Comm utativ e: A+B = B+ A
2
• Asso ciativ e: (A+B)+C = A+(B+C)
• Iden t it y: A+O = A , where O is the zero matrix of the same order as A
• In v erse: A+(-A) = O
3.2 Mu ltiplication of Matrices
F or matrices A = [a
ij
]
m×n
and B = [b
ij
]
n×p
, their pro duct C = AB is a matrix of order
m× p :
?
?
?
?
?
c
11
c
12
··· c
1p
c
21
c
22
··· c
2p
.
.
.
.
.
.
.
.
.
.
.
.
c
m1
c
m2
··· c
mp
?
?
?
?
?
where:
c
ij
=
n
?
k=1
a
ik
b
kj
, i = 1,2,...,m, j = 1,2,...,p
The pr op erties of matrix m ultiplication include:
• Non- comm utativ e: In general, AB ?= BA
• Asso c iativ e: (AB)C = A(BC)
• Dist ributiv e: A(B+C) = AB+ AC and (A+B)C = AC+BC
• Zer o divisors: If AB = O , it do es not necessarily imply A = O or B = O
4 Sp ecial T yp es of Matrices
Matrices can b e classified based on their prop erties. Belo w are the definitions along with
examples:
• Diago nal Matrix: A square matrix where a
ij
= 0 for i ?= j .
Example:
[
5 0
0 -2
]
• Symm etric Matrix: A square matrix A is symmetric if A = A
T
.
Example:
[
1 4
4 3
]
• Sk ew-Symmetric Matrix: A square matrix A is sk ew-symmetric if A = -A
T
. This
implies the diagonal elemen ts are zero.
Example:
[
0 2
-2 0
]
3
Page 3


1 Matrices
A matrix is a rectangular arra y of n um b ers, sym b ols, or expressions arranged in ro ws and
columns. If A = [a
ij
]
m×n
, then A is a matrix of order m×n , where:
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
Here, m is the n um b er of ro ws, n is the n um b er of columns, and a
ij
represen ts the elemen t
in the i -th ro w and j -th column.
2 Notation and Sp ecial Characters
Matrices often use sp ecial c haracters and notations to represen t their elemen ts and op er-
ations. Belo w is an explanation of the k ey sym b ols used in this do cumen t:
• a
ij
: The elemen t in thei -th ro w and j -th column of a matrix. The subscriptij indicates
the p osition, where i is the ro w index and j is the c olumn index.
• A
T
: The transp ose of matrix A . The sup erscript T denotes the op eration of in ter-
c hanging ro ws and columns.
• A
*
: The conjugate transp ose (or adjoin t) of matrix A . The sup erscript * indicates
that the matrix is first conjugated (complex conjugate of eac h elemen t) and then
transp osed.
• A : The conjugate of matrix A . The o v erline sym b ol denotes the op eration of taking
the complex conjugate of eac h elemen t.
• ? : The summation sym b ol, used in matrix m ultiplication and trace. F or example,
?
n
k=1
a
ik
b
kj
represen ts the sum of pro ducts for computing an elemen t in matrix m ulti-
plication.
• det(A) : The determinan t of matrix A . The notation det ? is a function that computes
a scalar v alue for a square matrix.
• |A| : An alternativ e notation for the determinan t of A , often used in the con text of
in v erses.
•
.
.
. : A sym b ol used in matrix notation to indicate that the pattern of elemen ts con-
tin ues diagonally (e.g., in a general m×n matrix).
3 Basic Op erations on Matrices
3.1 A ddition and Subtraction
Matrices can b e added or subtracted if they ha v e the same order. The follo wing prop erties
hold:
• Comm utativ e: A+B = B+ A
2
• Asso ciativ e: (A+B)+C = A+(B+C)
• Iden t it y: A+O = A , where O is the zero matrix of the same order as A
• In v erse: A+(-A) = O
3.2 Mu ltiplication of Matrices
F or matrices A = [a
ij
]
m×n
and B = [b
ij
]
n×p
, their pro duct C = AB is a matrix of order
m× p :
?
?
?
?
?
c
11
c
12
··· c
1p
c
21
c
22
··· c
2p
.
.
.
.
.
.
.
.
.
.
.
.
c
m1
c
m2
··· c
mp
?
?
?
?
?
where:
c
ij
=
n
?
k=1
a
ik
b
kj
, i = 1,2,...,m, j = 1,2,...,p
The pr op erties of matrix m ultiplication include:
• Non- comm utativ e: In general, AB ?= BA
• Asso c iativ e: (AB)C = A(BC)
• Dist ributiv e: A(B+C) = AB+ AC and (A+B)C = AC+BC
• Zer o divisors: If AB = O , it do es not necessarily imply A = O or B = O
4 Sp ecial T yp es of Matrices
Matrices can b e classified based on their prop erties. Belo w are the definitions along with
examples:
• Diago nal Matrix: A square matrix where a
ij
= 0 for i ?= j .
Example:
[
5 0
0 -2
]
• Symm etric Matrix: A square matrix A is symmetric if A = A
T
.
Example:
[
1 4
4 3
]
• Sk ew-Symmetric Matrix: A square matrix A is sk ew-symmetric if A = -A
T
. This
implies the diagonal elemen ts are zero.
Example:
[
0 2
-2 0
]
3
• Hermitian Matrix: A square matrix A is Hermitian if A = A
*
, where A
*
is the
conjugate transp ose. This means a
ij
= a
ji
.
Example:
[
3 2+i
2-i 5
]
• Sk ew-Hermitian Matrix: A square matrix A is sk ew-Hermitian if A = -A
*
. The
diagonal elemen ts are purely imaginary or zero.
Example:
[
i 2+i
-2+i -3i
]
• Orthogonal Matrix: A square matrix A is orthogonal if A
T
A = I .
Example:
[
cos? -sin?
sin? cos?
]
(e.g., for ? = 0 , this is
[
1 0
0 1
]
)
• Idemp oten t Matrix: A matrix A is idemp oten t if A
2
= A .
Example:
[
1 0
0 0
]
• In v olun tary Matrix: A matrix A is in v olun tary if A
2
= I .
Example:
[
0 1
1 0
]
• Nilp oten t Matrix: A matrix A is nilp oten t if there exists a p ositiv e in teger p suc h that
A
p
= O .
Example:
?
?
0 1 0
0 0 1
0 0 0
?
?
Let’s v erify: A
2
=
?
?
0 0 1
0 0 0
0 0 0
?
?
, and A
3
= A
2
· A =
?
?
0 0 0
0 0 0
0 0 0
?
?
= O , so p = 3 .
5 T race of a Matrix
The trace of a square matrix A = [a
ij
]
n×n
, denoted tr(A) , is the sum of its diagonal
elemen ts:
tr(A) =
n
?
i=1
a
ii
= a
11
+a
22
+···+a
nn
Here are some examples:
• Example 1 (2x2 matrix): Consider the matrix
A =
[
4 1
3 -2
]
The trace is:
tr(A) = 4+(-2) = 2
4
Page 4


1 Matrices
A matrix is a rectangular arra y of n um b ers, sym b ols, or expressions arranged in ro ws and
columns. If A = [a
ij
]
m×n
, then A is a matrix of order m×n , where:
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
Here, m is the n um b er of ro ws, n is the n um b er of columns, and a
ij
represen ts the elemen t
in the i -th ro w and j -th column.
2 Notation and Sp ecial Characters
Matrices often use sp ecial c haracters and notations to represen t their elemen ts and op er-
ations. Belo w is an explanation of the k ey sym b ols used in this do cumen t:
• a
ij
: The elemen t in thei -th ro w and j -th column of a matrix. The subscriptij indicates
the p osition, where i is the ro w index and j is the c olumn index.
• A
T
: The transp ose of matrix A . The sup erscript T denotes the op eration of in ter-
c hanging ro ws and columns.
• A
*
: The conjugate transp ose (or adjoin t) of matrix A . The sup erscript * indicates
that the matrix is first conjugated (complex conjugate of eac h elemen t) and then
transp osed.
• A : The conjugate of matrix A . The o v erline sym b ol denotes the op eration of taking
the complex conjugate of eac h elemen t.
• ? : The summation sym b ol, used in matrix m ultiplication and trace. F or example,
?
n
k=1
a
ik
b
kj
represen ts the sum of pro ducts for computing an elemen t in matrix m ulti-
plication.
• det(A) : The determinan t of matrix A . The notation det ? is a function that computes
a scalar v alue for a square matrix.
• |A| : An alternativ e notation for the determinan t of A , often used in the con text of
in v erses.
•
.
.
. : A sym b ol used in matrix notation to indicate that the pattern of elemen ts con-
tin ues diagonally (e.g., in a general m×n matrix).
3 Basic Op erations on Matrices
3.1 A ddition and Subtraction
Matrices can b e added or subtracted if they ha v e the same order. The follo wing prop erties
hold:
• Comm utativ e: A+B = B+ A
2
• Asso ciativ e: (A+B)+C = A+(B+C)
• Iden t it y: A+O = A , where O is the zero matrix of the same order as A
• In v erse: A+(-A) = O
3.2 Mu ltiplication of Matrices
F or matrices A = [a
ij
]
m×n
and B = [b
ij
]
n×p
, their pro duct C = AB is a matrix of order
m× p :
?
?
?
?
?
c
11
c
12
··· c
1p
c
21
c
22
··· c
2p
.
.
.
.
.
.
.
.
.
.
.
.
c
m1
c
m2
··· c
mp
?
?
?
?
?
where:
c
ij
=
n
?
k=1
a
ik
b
kj
, i = 1,2,...,m, j = 1,2,...,p
The pr op erties of matrix m ultiplication include:
• Non- comm utativ e: In general, AB ?= BA
• Asso c iativ e: (AB)C = A(BC)
• Dist ributiv e: A(B+C) = AB+ AC and (A+B)C = AC+BC
• Zer o divisors: If AB = O , it do es not necessarily imply A = O or B = O
4 Sp ecial T yp es of Matrices
Matrices can b e classified based on their prop erties. Belo w are the definitions along with
examples:
• Diago nal Matrix: A square matrix where a
ij
= 0 for i ?= j .
Example:
[
5 0
0 -2
]
• Symm etric Matrix: A square matrix A is symmetric if A = A
T
.
Example:
[
1 4
4 3
]
• Sk ew-Symmetric Matrix: A square matrix A is sk ew-symmetric if A = -A
T
. This
implies the diagonal elemen ts are zero.
Example:
[
0 2
-2 0
]
3
• Hermitian Matrix: A square matrix A is Hermitian if A = A
*
, where A
*
is the
conjugate transp ose. This means a
ij
= a
ji
.
Example:
[
3 2+i
2-i 5
]
• Sk ew-Hermitian Matrix: A square matrix A is sk ew-Hermitian if A = -A
*
. The
diagonal elemen ts are purely imaginary or zero.
Example:
[
i 2+i
-2+i -3i
]
• Orthogonal Matrix: A square matrix A is orthogonal if A
T
A = I .
Example:
[
cos? -sin?
sin? cos?
]
(e.g., for ? = 0 , this is
[
1 0
0 1
]
)
• Idemp oten t Matrix: A matrix A is idemp oten t if A
2
= A .
Example:
[
1 0
0 0
]
• In v olun tary Matrix: A matrix A is in v olun tary if A
2
= I .
Example:
[
0 1
1 0
]
• Nilp oten t Matrix: A matrix A is nilp oten t if there exists a p ositiv e in teger p suc h that
A
p
= O .
Example:
?
?
0 1 0
0 0 1
0 0 0
?
?
Let’s v erify: A
2
=
?
?
0 0 1
0 0 0
0 0 0
?
?
, and A
3
= A
2
· A =
?
?
0 0 0
0 0 0
0 0 0
?
?
= O , so p = 3 .
5 T race of a Matrix
The trace of a square matrix A = [a
ij
]
n×n
, denoted tr(A) , is the sum of its diagonal
elemen ts:
tr(A) =
n
?
i=1
a
ii
= a
11
+a
22
+···+a
nn
Here are some examples:
• Example 1 (2x2 matrix): Consider the matrix
A =
[
4 1
3 -2
]
The trace is:
tr(A) = 4+(-2) = 2
4
• Example 2 (3x3 matrix): Consider the matrix
B =
?
?
1 0 2
3 5 4
0 1 -1
?
?
The trace is:
tr(B) = 1+5+(-1) = 5
Prop erties:
• t r(A) = tr(A
T
)
• t r(AB) = tr(BA)
6 T ransp ose of a Matrix
The transp ose of a matrix A = [a
ij
]
m×n
, denoted A
T
, is obtained b y in terc hanging its
ro ws and columns, resulting in A
T
= [a
ji
]
n×m
. Here are some examples:
• Example 1 (2x3 matrix): Consider the matrix
A =
[
1 2 3
4 5 6
]
The transp ose is:
A
T
=
?
?
1 4
2 5
3 6
?
?
• Example 2 (3x3 matrix): Consider the matrix
B =
?
?
1 2 3
4 5 6
7 8 9
?
?
The transp ose is:
B
T
=
?
?
1 4 7
2 5 8
3 6 9
?
?
Prop erties:
• (A
T
)
T
= A
• (AB)
T
= B
T
A
T
• (kA)
T
= kA
T
• (A+B)
T
= A
T
+B
T
• I
T
= I
• tr(A) = tr(A
T
)
5
Page 5


1 Matrices
A matrix is a rectangular arra y of n um b ers, sym b ols, or expressions arranged in ro ws and
columns. If A = [a
ij
]
m×n
, then A is a matrix of order m×n , where:
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
Here, m is the n um b er of ro ws, n is the n um b er of columns, and a
ij
represen ts the elemen t
in the i -th ro w and j -th column.
2 Notation and Sp ecial Characters
Matrices often use sp ecial c haracters and notations to represen t their elemen ts and op er-
ations. Belo w is an explanation of the k ey sym b ols used in this do cumen t:
• a
ij
: The elemen t in thei -th ro w and j -th column of a matrix. The subscriptij indicates
the p osition, where i is the ro w index and j is the c olumn index.
• A
T
: The transp ose of matrix A . The sup erscript T denotes the op eration of in ter-
c hanging ro ws and columns.
• A
*
: The conjugate transp ose (or adjoin t) of matrix A . The sup erscript * indicates
that the matrix is first conjugated (complex conjugate of eac h elemen t) and then
transp osed.
• A : The conjugate of matrix A . The o v erline sym b ol denotes the op eration of taking
the complex conjugate of eac h elemen t.
• ? : The summation sym b ol, used in matrix m ultiplication and trace. F or example,
?
n
k=1
a
ik
b
kj
represen ts the sum of pro ducts for computing an elemen t in matrix m ulti-
plication.
• det(A) : The determinan t of matrix A . The notation det ? is a function that computes
a scalar v alue for a square matrix.
• |A| : An alternativ e notation for the determinan t of A , often used in the con text of
in v erses.
•
.
.
. : A sym b ol used in matrix notation to indicate that the pattern of elemen ts con-
tin ues diagonally (e.g., in a general m×n matrix).
3 Basic Op erations on Matrices
3.1 A ddition and Subtraction
Matrices can b e added or subtracted if they ha v e the same order. The follo wing prop erties
hold:
• Comm utativ e: A+B = B+ A
2
• Asso ciativ e: (A+B)+C = A+(B+C)
• Iden t it y: A+O = A , where O is the zero matrix of the same order as A
• In v erse: A+(-A) = O
3.2 Mu ltiplication of Matrices
F or matrices A = [a
ij
]
m×n
and B = [b
ij
]
n×p
, their pro duct C = AB is a matrix of order
m× p :
?
?
?
?
?
c
11
c
12
··· c
1p
c
21
c
22
··· c
2p
.
.
.
.
.
.
.
.
.
.
.
.
c
m1
c
m2
··· c
mp
?
?
?
?
?
where:
c
ij
=
n
?
k=1
a
ik
b
kj
, i = 1,2,...,m, j = 1,2,...,p
The pr op erties of matrix m ultiplication include:
• Non- comm utativ e: In general, AB ?= BA
• Asso c iativ e: (AB)C = A(BC)
• Dist ributiv e: A(B+C) = AB+ AC and (A+B)C = AC+BC
• Zer o divisors: If AB = O , it do es not necessarily imply A = O or B = O
4 Sp ecial T yp es of Matrices
Matrices can b e classified based on their prop erties. Belo w are the definitions along with
examples:
• Diago nal Matrix: A square matrix where a
ij
= 0 for i ?= j .
Example:
[
5 0
0 -2
]
• Symm etric Matrix: A square matrix A is symmetric if A = A
T
.
Example:
[
1 4
4 3
]
• Sk ew-Symmetric Matrix: A square matrix A is sk ew-symmetric if A = -A
T
. This
implies the diagonal elemen ts are zero.
Example:
[
0 2
-2 0
]
3
• Hermitian Matrix: A square matrix A is Hermitian if A = A
*
, where A
*
is the
conjugate transp ose. This means a
ij
= a
ji
.
Example:
[
3 2+i
2-i 5
]
• Sk ew-Hermitian Matrix: A square matrix A is sk ew-Hermitian if A = -A
*
. The
diagonal elemen ts are purely imaginary or zero.
Example:
[
i 2+i
-2+i -3i
]
• Orthogonal Matrix: A square matrix A is orthogonal if A
T
A = I .
Example:
[
cos? -sin?
sin? cos?
]
(e.g., for ? = 0 , this is
[
1 0
0 1
]
)
• Idemp oten t Matrix: A matrix A is idemp oten t if A
2
= A .
Example:
[
1 0
0 0
]
• In v olun tary Matrix: A matrix A is in v olun tary if A
2
= I .
Example:
[
0 1
1 0
]
• Nilp oten t Matrix: A matrix A is nilp oten t if there exists a p ositiv e in teger p suc h that
A
p
= O .
Example:
?
?
0 1 0
0 0 1
0 0 0
?
?
Let’s v erify: A
2
=
?
?
0 0 1
0 0 0
0 0 0
?
?
, and A
3
= A
2
· A =
?
?
0 0 0
0 0 0
0 0 0
?
?
= O , so p = 3 .
5 T race of a Matrix
The trace of a square matrix A = [a
ij
]
n×n
, denoted tr(A) , is the sum of its diagonal
elemen ts:
tr(A) =
n
?
i=1
a
ii
= a
11
+a
22
+···+a
nn
Here are some examples:
• Example 1 (2x2 matrix): Consider the matrix
A =
[
4 1
3 -2
]
The trace is:
tr(A) = 4+(-2) = 2
4
• Example 2 (3x3 matrix): Consider the matrix
B =
?
?
1 0 2
3 5 4
0 1 -1
?
?
The trace is:
tr(B) = 1+5+(-1) = 5
Prop erties:
• t r(A) = tr(A
T
)
• t r(AB) = tr(BA)
6 T ransp ose of a Matrix
The transp ose of a matrix A = [a
ij
]
m×n
, denoted A
T
, is obtained b y in terc hanging its
ro ws and columns, resulting in A
T
= [a
ji
]
n×m
. Here are some examples:
• Example 1 (2x3 matrix): Consider the matrix
A =
[
1 2 3
4 5 6
]
The transp ose is:
A
T
=
?
?
1 4
2 5
3 6
?
?
• Example 2 (3x3 matrix): Consider the matrix
B =
?
?
1 2 3
4 5 6
7 8 9
?
?
The transp ose is:
B
T
=
?
?
1 4 7
2 5 8
3 6 9
?
?
Prop erties:
• (A
T
)
T
= A
• (AB)
T
= B
T
A
T
• (kA)
T
= kA
T
• (A+B)
T
= A
T
+B
T
• I
T
= I
• tr(A) = tr(A
T
)
5
7 Conjugate and T ransp ose Conjugate of a Matrix
The conjugate of a matrix A , denoted A , is obtained b y taking the complex conjugate of
eac h elemen t. F or example, if an elemen t a
ij
= x+yi , then a
ij
= x-yi . Prop erties:
• A = A (taking the conjugate t wice returns the original matrix)
• A+B = A+B
• kA = kA
• AB = AB
The transp ose conjugate (or adjoin t) of a matrix A , denoted A
*
, is A
*
= (A)
T
. Prop er-
ties:
• (A
*
)
*
= A (since A
*
= (A)
T
, applying the op eration t wice returns A )
• (A+B)
*
= A
*
+B
*
• (kA)
*
= kA
*
• (AB)
*
= B
*
A
*
8 A d join t of a Matrix
F o r a square matrix A , the adjoin t, denoted adj(A) , is the transp ose of the cofactor
matrix of A . Prop erties:
• adj(A) = adj(A
T
)
• adj(AB) = adj(B) adj(A)
• adj(kA) = k
n-1
adj(A) , for an n×n matrix
• adj(A
T
) = [ adj(A)]
T
9 In v erse of a Matrix
A square matrix A has an in v erse, denoted A
-1
, if AB = BA = I . The in v erse exists if
|A| ?= 0 :
A
-1
=
1
|A|
adj(A)
Prop erties:
• (A
-1
)
-1
= A
• (AB)
-1
= B
-1
A
-1
• (A
T
)
-1
= (A
-1
)
T
6