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Page 1
Matrices
4.1 Introduction
A rectangular array of ?? * ?? numbers consisting of ?? rows and ?? columns is termed as a matrix
of order ?? × ?? and given as:
?? = (
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
) or ?? = [
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
]
It may also be denoted as ?? = [?? ????
], ?? = 1…?? , ?? = 1…??
Null Matrix: A matrix with all zero elements is known as a null matrix or zero matrix.
Square matrix: A matrix having equal number of rows and columns is called a square matrix.
?? = (
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
) is a square matrix of order ?? × ??
Sum of all elements in the principal diagonal of a square matrix ?? is known as ‘Trace ?? ’
or ‘Spur ?? ’. ? Trace ?? = ?? ????
+ ?? ????
+ ?+ ?? ????
Identity or Unit Matrix: A square matrix having all principal diagonal elements unity and non-
diagonal elements zero is called an identity matrix.
?? = (
1 0 0
0 1 0
0 0 1
) is an identity matrix of order 3
Triangular Matrix: A square matrix in which all elements above or below principal diagonal are
zero is called a triangular matrix.
?? = (
3 0 0
4 2 0
2 6 1
) ?? = (
3 5 2
0 2 6
0 0 1
)
Lower Triangular Matrix Upper Triangular Matrix
Diagonal Matrix: A square matrix having all non-diagonal elements zero is called a diagonal
matrix.
?? = (
3 0 0
0 4 0
0 0 2
) is a diagonal matrix of order 3
Scalar Matrix: A diagonal matrix with all equal elements is called a scalar matrix.
Page 2
Matrices
4.1 Introduction
A rectangular array of ?? * ?? numbers consisting of ?? rows and ?? columns is termed as a matrix
of order ?? × ?? and given as:
?? = (
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
) or ?? = [
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
]
It may also be denoted as ?? = [?? ????
], ?? = 1…?? , ?? = 1…??
Null Matrix: A matrix with all zero elements is known as a null matrix or zero matrix.
Square matrix: A matrix having equal number of rows and columns is called a square matrix.
?? = (
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
) is a square matrix of order ?? × ??
Sum of all elements in the principal diagonal of a square matrix ?? is known as ‘Trace ?? ’
or ‘Spur ?? ’. ? Trace ?? = ?? ????
+ ?? ????
+ ?+ ?? ????
Identity or Unit Matrix: A square matrix having all principal diagonal elements unity and non-
diagonal elements zero is called an identity matrix.
?? = (
1 0 0
0 1 0
0 0 1
) is an identity matrix of order 3
Triangular Matrix: A square matrix in which all elements above or below principal diagonal are
zero is called a triangular matrix.
?? = (
3 0 0
4 2 0
2 6 1
) ?? = (
3 5 2
0 2 6
0 0 1
)
Lower Triangular Matrix Upper Triangular Matrix
Diagonal Matrix: A square matrix having all non-diagonal elements zero is called a diagonal
matrix.
?? = (
3 0 0
0 4 0
0 0 2
) is a diagonal matrix of order 3
Scalar Matrix: A diagonal matrix with all equal elements is called a scalar matrix.
?? = (
3 0 0
0 3 0
0 0 3
) is a scalar matrix of order 3
Singular Matrix: If the determinant of a square matrix is zero i.e., |?? | = 0 , then it is known as
a singular matrix.
?? = (
1 -1 0
0 1 -3
-2 1 3
) is a singular matrix of order 3
Transpose: The matrix ?? '
or ?? ?? obtained by interchanging rows and columns of a matrix ?? is
known as its transpose.
?? = (
1 3 5
2 -1 4
0 2 3
) ?? ?? = (
1 2 0
3 -1 2
5 4 3
)
Symmetric and Skew-Symmetric Matrices:
A square matrix ?? = [?? ????
] is said to be symmetric if ?? ?? = ?? or ?? ????
= ?? ????
? ?? ,?? and skew-
symmetric if ?? ?? = -?? or ?? ????
= -?? ????
? ?? ,??
?? = (
1 2 3
2 2 4
3 4 3
) ?? = (
0 -1 2
1 0 -3
-2 3 0
)
Symmetric Matrix Skew- Symmetric Matrix
Results: 1. Diagonal elements of a skew-symmetric matrix are all zero as
?? ????
= -?? ????
? ?? ????
= 0
2. Any real matrix can be uniquely expressed as the sum of a symmetric and a skew-
symmetric matrix as ?? =
1
2
( ?? + ?? ?? )+
1
2
( ?? - ?? ?? ) , where ( ?? + ?? ?? ) is symmetric, while
( ?? - ?? ?? ) is skew-symmetric
Orthogonal Matrix
A square matrix ?? = [?? ????
] is said to be orthogonal if ????
?? = ?? = ?? ?? ??
Result: If ?? and ?? are two orthogonal matrices, then ???? is also a orthogonal matrix.
Proof: ( ???? ) ( ???? )
?? = ( ???? ) ?? ?? ?? ?? ? ( ???? )
?? = ?? ?? ?? ??
= ?? ( ?? ?? ?? ) ?? ??
= ???? ?? ?? ? ?? is an orthogonal matrix
= ?? ?? ?? = ?? ? ?? is an orthogonal matrix
4.2 Algebra of Matrices
Addition and Subtraction of Matrix: Addition or subtraction can be performed on two matrices
if and only if they are of same order.
Page 3
Matrices
4.1 Introduction
A rectangular array of ?? * ?? numbers consisting of ?? rows and ?? columns is termed as a matrix
of order ?? × ?? and given as:
?? = (
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
) or ?? = [
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
]
It may also be denoted as ?? = [?? ????
], ?? = 1…?? , ?? = 1…??
Null Matrix: A matrix with all zero elements is known as a null matrix or zero matrix.
Square matrix: A matrix having equal number of rows and columns is called a square matrix.
?? = (
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
) is a square matrix of order ?? × ??
Sum of all elements in the principal diagonal of a square matrix ?? is known as ‘Trace ?? ’
or ‘Spur ?? ’. ? Trace ?? = ?? ????
+ ?? ????
+ ?+ ?? ????
Identity or Unit Matrix: A square matrix having all principal diagonal elements unity and non-
diagonal elements zero is called an identity matrix.
?? = (
1 0 0
0 1 0
0 0 1
) is an identity matrix of order 3
Triangular Matrix: A square matrix in which all elements above or below principal diagonal are
zero is called a triangular matrix.
?? = (
3 0 0
4 2 0
2 6 1
) ?? = (
3 5 2
0 2 6
0 0 1
)
Lower Triangular Matrix Upper Triangular Matrix
Diagonal Matrix: A square matrix having all non-diagonal elements zero is called a diagonal
matrix.
?? = (
3 0 0
0 4 0
0 0 2
) is a diagonal matrix of order 3
Scalar Matrix: A diagonal matrix with all equal elements is called a scalar matrix.
?? = (
3 0 0
0 3 0
0 0 3
) is a scalar matrix of order 3
Singular Matrix: If the determinant of a square matrix is zero i.e., |?? | = 0 , then it is known as
a singular matrix.
?? = (
1 -1 0
0 1 -3
-2 1 3
) is a singular matrix of order 3
Transpose: The matrix ?? '
or ?? ?? obtained by interchanging rows and columns of a matrix ?? is
known as its transpose.
?? = (
1 3 5
2 -1 4
0 2 3
) ?? ?? = (
1 2 0
3 -1 2
5 4 3
)
Symmetric and Skew-Symmetric Matrices:
A square matrix ?? = [?? ????
] is said to be symmetric if ?? ?? = ?? or ?? ????
= ?? ????
? ?? ,?? and skew-
symmetric if ?? ?? = -?? or ?? ????
= -?? ????
? ?? ,??
?? = (
1 2 3
2 2 4
3 4 3
) ?? = (
0 -1 2
1 0 -3
-2 3 0
)
Symmetric Matrix Skew- Symmetric Matrix
Results: 1. Diagonal elements of a skew-symmetric matrix are all zero as
?? ????
= -?? ????
? ?? ????
= 0
2. Any real matrix can be uniquely expressed as the sum of a symmetric and a skew-
symmetric matrix as ?? =
1
2
( ?? + ?? ?? )+
1
2
( ?? - ?? ?? ) , where ( ?? + ?? ?? ) is symmetric, while
( ?? - ?? ?? ) is skew-symmetric
Orthogonal Matrix
A square matrix ?? = [?? ????
] is said to be orthogonal if ????
?? = ?? = ?? ?? ??
Result: If ?? and ?? are two orthogonal matrices, then ???? is also a orthogonal matrix.
Proof: ( ???? ) ( ???? )
?? = ( ???? ) ?? ?? ?? ?? ? ( ???? )
?? = ?? ?? ?? ??
= ?? ( ?? ?? ?? ) ?? ??
= ???? ?? ?? ? ?? is an orthogonal matrix
= ?? ?? ?? = ?? ? ?? is an orthogonal matrix
4.2 Algebra of Matrices
Addition and Subtraction of Matrix: Addition or subtraction can be performed on two matrices
if and only if they are of same order.
?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
) ?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
)
Then ?? ± ?? = (
?? 11
± ?? 11
?? 12
± ?? 12
?? 13
± ?? 13
?? 21
± ?? 21
?? 22
± ?? 22
?? 23
± ?? 23
?? 31
± ?? 31
?? 32
± ?? 32
?? 33
± ?? 33
)
Multiplication of Matrix by a Scalar: If we multiply a matrix ?? by a scalar ?? , then each element
of the matrix is multiplied by ??
?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
) ???? = (
????
11
?? ?? 12
?? ?? 13
?? ?? 21
????
22
?? ?? 23
?? ?? 31
?? ?? 32
????
33
)
Multiplication of Two Matrices: Matrix product ???? is possible only if number of columns in
matrix ?? are same as number of rows in matrix ?? .
?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
) ?? = (
?? 11
?? 12
?? 21
?? 22
?? 31
?? 32
)
?? = ???? = (
?? 11
= ?? 11
?? 11
+ ?? 12
?? 21
+ ?? 13
?? 31
?? 12
= ?? 11
?? 12
+ ?? 12
?? 22
+ ?? 13
?? 32
?? 21
= ?? 21
?? 11
+ ?? 22
?? 21
+ ?? 23
?? 31
?? 22
= ?? 21
?? 12
+ ?? 22
?? 22
+ ?? 23
?? 32
)
Note that: (i) ?? ?? ×?? ?? ?? ×?? = ?? ?? ×??
(ii) ???? ? ???? in general
(iii) ???? = 0 does not necessarily imply that ?? = 0 or ?? = 0
(iv) ???? = 0 does not necessarily imply that ???? = 0
For example, ?? = (
0 1
0 0
) ?? = (
1 0
0 0
) ???? = (
0 0
0 0
) ???? = (
0 1
0 0
)
Example 1 If ?? = (
sin?? cos?? sin?? cos?? ) ?? = (
sin?? sin?? cos?? cos ?? ) find ???? and ????
Solution: ???? = (
sin
2
?? + cos
2
?? sin
2
?? + cos
2
??
sin
2
?? + cos
2
?? sin
2
?? + cos
2
?? )= (
1 1
1 1
)
???? = (
2sin
2
?? sin2?? sin2?? 2cos
2
?? )
Example 2 Express the matrix ?? = (
1 3 5
2 -1 4
0 2 3
) as the sum of symmetric and skew-
symmetric matrices.
Solution: ?? = (
1 3 5
2 -1 4
1 2 3
) ?? ?? = (
1 2 1
3 -1 2
5 4 3
)
Page 4
Matrices
4.1 Introduction
A rectangular array of ?? * ?? numbers consisting of ?? rows and ?? columns is termed as a matrix
of order ?? × ?? and given as:
?? = (
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
) or ?? = [
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
]
It may also be denoted as ?? = [?? ????
], ?? = 1…?? , ?? = 1…??
Null Matrix: A matrix with all zero elements is known as a null matrix or zero matrix.
Square matrix: A matrix having equal number of rows and columns is called a square matrix.
?? = (
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
) is a square matrix of order ?? × ??
Sum of all elements in the principal diagonal of a square matrix ?? is known as ‘Trace ?? ’
or ‘Spur ?? ’. ? Trace ?? = ?? ????
+ ?? ????
+ ?+ ?? ????
Identity or Unit Matrix: A square matrix having all principal diagonal elements unity and non-
diagonal elements zero is called an identity matrix.
?? = (
1 0 0
0 1 0
0 0 1
) is an identity matrix of order 3
Triangular Matrix: A square matrix in which all elements above or below principal diagonal are
zero is called a triangular matrix.
?? = (
3 0 0
4 2 0
2 6 1
) ?? = (
3 5 2
0 2 6
0 0 1
)
Lower Triangular Matrix Upper Triangular Matrix
Diagonal Matrix: A square matrix having all non-diagonal elements zero is called a diagonal
matrix.
?? = (
3 0 0
0 4 0
0 0 2
) is a diagonal matrix of order 3
Scalar Matrix: A diagonal matrix with all equal elements is called a scalar matrix.
?? = (
3 0 0
0 3 0
0 0 3
) is a scalar matrix of order 3
Singular Matrix: If the determinant of a square matrix is zero i.e., |?? | = 0 , then it is known as
a singular matrix.
?? = (
1 -1 0
0 1 -3
-2 1 3
) is a singular matrix of order 3
Transpose: The matrix ?? '
or ?? ?? obtained by interchanging rows and columns of a matrix ?? is
known as its transpose.
?? = (
1 3 5
2 -1 4
0 2 3
) ?? ?? = (
1 2 0
3 -1 2
5 4 3
)
Symmetric and Skew-Symmetric Matrices:
A square matrix ?? = [?? ????
] is said to be symmetric if ?? ?? = ?? or ?? ????
= ?? ????
? ?? ,?? and skew-
symmetric if ?? ?? = -?? or ?? ????
= -?? ????
? ?? ,??
?? = (
1 2 3
2 2 4
3 4 3
) ?? = (
0 -1 2
1 0 -3
-2 3 0
)
Symmetric Matrix Skew- Symmetric Matrix
Results: 1. Diagonal elements of a skew-symmetric matrix are all zero as
?? ????
= -?? ????
? ?? ????
= 0
2. Any real matrix can be uniquely expressed as the sum of a symmetric and a skew-
symmetric matrix as ?? =
1
2
( ?? + ?? ?? )+
1
2
( ?? - ?? ?? ) , where ( ?? + ?? ?? ) is symmetric, while
( ?? - ?? ?? ) is skew-symmetric
Orthogonal Matrix
A square matrix ?? = [?? ????
] is said to be orthogonal if ????
?? = ?? = ?? ?? ??
Result: If ?? and ?? are two orthogonal matrices, then ???? is also a orthogonal matrix.
Proof: ( ???? ) ( ???? )
?? = ( ???? ) ?? ?? ?? ?? ? ( ???? )
?? = ?? ?? ?? ??
= ?? ( ?? ?? ?? ) ?? ??
= ???? ?? ?? ? ?? is an orthogonal matrix
= ?? ?? ?? = ?? ? ?? is an orthogonal matrix
4.2 Algebra of Matrices
Addition and Subtraction of Matrix: Addition or subtraction can be performed on two matrices
if and only if they are of same order.
?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
) ?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
)
Then ?? ± ?? = (
?? 11
± ?? 11
?? 12
± ?? 12
?? 13
± ?? 13
?? 21
± ?? 21
?? 22
± ?? 22
?? 23
± ?? 23
?? 31
± ?? 31
?? 32
± ?? 32
?? 33
± ?? 33
)
Multiplication of Matrix by a Scalar: If we multiply a matrix ?? by a scalar ?? , then each element
of the matrix is multiplied by ??
?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
) ???? = (
????
11
?? ?? 12
?? ?? 13
?? ?? 21
????
22
?? ?? 23
?? ?? 31
?? ?? 32
????
33
)
Multiplication of Two Matrices: Matrix product ???? is possible only if number of columns in
matrix ?? are same as number of rows in matrix ?? .
?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
) ?? = (
?? 11
?? 12
?? 21
?? 22
?? 31
?? 32
)
?? = ???? = (
?? 11
= ?? 11
?? 11
+ ?? 12
?? 21
+ ?? 13
?? 31
?? 12
= ?? 11
?? 12
+ ?? 12
?? 22
+ ?? 13
?? 32
?? 21
= ?? 21
?? 11
+ ?? 22
?? 21
+ ?? 23
?? 31
?? 22
= ?? 21
?? 12
+ ?? 22
?? 22
+ ?? 23
?? 32
)
Note that: (i) ?? ?? ×?? ?? ?? ×?? = ?? ?? ×??
(ii) ???? ? ???? in general
(iii) ???? = 0 does not necessarily imply that ?? = 0 or ?? = 0
(iv) ???? = 0 does not necessarily imply that ???? = 0
For example, ?? = (
0 1
0 0
) ?? = (
1 0
0 0
) ???? = (
0 0
0 0
) ???? = (
0 1
0 0
)
Example 1 If ?? = (
sin?? cos?? sin?? cos?? ) ?? = (
sin?? sin?? cos?? cos ?? ) find ???? and ????
Solution: ???? = (
sin
2
?? + cos
2
?? sin
2
?? + cos
2
??
sin
2
?? + cos
2
?? sin
2
?? + cos
2
?? )= (
1 1
1 1
)
???? = (
2sin
2
?? sin2?? sin2?? 2cos
2
?? )
Example 2 Express the matrix ?? = (
1 3 5
2 -1 4
0 2 3
) as the sum of symmetric and skew-
symmetric matrices.
Solution: ?? = (
1 3 5
2 -1 4
1 2 3
) ?? ?? = (
1 2 1
3 -1 2
5 4 3
)
1
2
( ?? + ?? ?? )= (
1
5
2
3
5
2
-1 3
3 3 3
) ,
1
2
( ?? - ?? ?? )= (
0
1
2
2
-1
2
0 1
-2 -1 0
)
? ?? = (
1
5
2
3
5
2
-1 3
3 3 3
) + (
0
1
2
2
-1
2
0 1
-2 -1 0
)
: Symmetric Skew-Symmetric
4.3 Minors, Cofactors, Determinants and Adjoint of a matrix
Minors associated with elements of a square matrix
A minor of each element of a square matrix is the unique value of the determinant associated with
it, which is obtained after eliminating the row and column in which the element exists.
For a 2 × 2 matrix ?? = (
?? 11
?? 12
?? 21
?? 22
)
?? 11
= ?? 22
, ?? 12
= ?? 21
, ?? 21
= ?? 12
, ?? 22
= ?? 11
For a 3 × 3 matrix ?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
)
?? 11
= |
?? 22
?? 23
?? 32
?? 33
| , ?? 12
= |
?? 21
?? 23
?? 31
?? 33
| ,…, ?? 33
= |
?? 11
?? 12
?? 21
?? 22
|
Cofactors associated with elements of a square matrix
The cofactor of each element is obtained on multiplying its minor by ( -1)
?? +?? .
?? ????
= ( -1)
?? +?? ?? ????
Determinant of a square matrix
Every square matrix is associated with a determinant and is denoted by det ( ?? ) or |?? |.
det ( ?? )= |?? | = |
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
|
Determinant of order ?? can be expanded by any one row or column using the formula
|?? | = ? ?? ????
?? ?? =1
?? ????
, where ?? ????
is the cofactor corresponding to the element ?? ????
.
A determinant of order 2 is evaluated as:
|?? | = |
?? 11
?? 12
?? 21
?? 22
| = ?? 11
?? 22
- ?? 12
?? 21
A determinant of order 3 is evaluated as:
Page 5
Matrices
4.1 Introduction
A rectangular array of ?? * ?? numbers consisting of ?? rows and ?? columns is termed as a matrix
of order ?? × ?? and given as:
?? = (
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
) or ?? = [
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
]
It may also be denoted as ?? = [?? ????
], ?? = 1…?? , ?? = 1…??
Null Matrix: A matrix with all zero elements is known as a null matrix or zero matrix.
Square matrix: A matrix having equal number of rows and columns is called a square matrix.
?? = (
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
) is a square matrix of order ?? × ??
Sum of all elements in the principal diagonal of a square matrix ?? is known as ‘Trace ?? ’
or ‘Spur ?? ’. ? Trace ?? = ?? ????
+ ?? ????
+ ?+ ?? ????
Identity or Unit Matrix: A square matrix having all principal diagonal elements unity and non-
diagonal elements zero is called an identity matrix.
?? = (
1 0 0
0 1 0
0 0 1
) is an identity matrix of order 3
Triangular Matrix: A square matrix in which all elements above or below principal diagonal are
zero is called a triangular matrix.
?? = (
3 0 0
4 2 0
2 6 1
) ?? = (
3 5 2
0 2 6
0 0 1
)
Lower Triangular Matrix Upper Triangular Matrix
Diagonal Matrix: A square matrix having all non-diagonal elements zero is called a diagonal
matrix.
?? = (
3 0 0
0 4 0
0 0 2
) is a diagonal matrix of order 3
Scalar Matrix: A diagonal matrix with all equal elements is called a scalar matrix.
?? = (
3 0 0
0 3 0
0 0 3
) is a scalar matrix of order 3
Singular Matrix: If the determinant of a square matrix is zero i.e., |?? | = 0 , then it is known as
a singular matrix.
?? = (
1 -1 0
0 1 -3
-2 1 3
) is a singular matrix of order 3
Transpose: The matrix ?? '
or ?? ?? obtained by interchanging rows and columns of a matrix ?? is
known as its transpose.
?? = (
1 3 5
2 -1 4
0 2 3
) ?? ?? = (
1 2 0
3 -1 2
5 4 3
)
Symmetric and Skew-Symmetric Matrices:
A square matrix ?? = [?? ????
] is said to be symmetric if ?? ?? = ?? or ?? ????
= ?? ????
? ?? ,?? and skew-
symmetric if ?? ?? = -?? or ?? ????
= -?? ????
? ?? ,??
?? = (
1 2 3
2 2 4
3 4 3
) ?? = (
0 -1 2
1 0 -3
-2 3 0
)
Symmetric Matrix Skew- Symmetric Matrix
Results: 1. Diagonal elements of a skew-symmetric matrix are all zero as
?? ????
= -?? ????
? ?? ????
= 0
2. Any real matrix can be uniquely expressed as the sum of a symmetric and a skew-
symmetric matrix as ?? =
1
2
( ?? + ?? ?? )+
1
2
( ?? - ?? ?? ) , where ( ?? + ?? ?? ) is symmetric, while
( ?? - ?? ?? ) is skew-symmetric
Orthogonal Matrix
A square matrix ?? = [?? ????
] is said to be orthogonal if ????
?? = ?? = ?? ?? ??
Result: If ?? and ?? are two orthogonal matrices, then ???? is also a orthogonal matrix.
Proof: ( ???? ) ( ???? )
?? = ( ???? ) ?? ?? ?? ?? ? ( ???? )
?? = ?? ?? ?? ??
= ?? ( ?? ?? ?? ) ?? ??
= ???? ?? ?? ? ?? is an orthogonal matrix
= ?? ?? ?? = ?? ? ?? is an orthogonal matrix
4.2 Algebra of Matrices
Addition and Subtraction of Matrix: Addition or subtraction can be performed on two matrices
if and only if they are of same order.
?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
) ?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
)
Then ?? ± ?? = (
?? 11
± ?? 11
?? 12
± ?? 12
?? 13
± ?? 13
?? 21
± ?? 21
?? 22
± ?? 22
?? 23
± ?? 23
?? 31
± ?? 31
?? 32
± ?? 32
?? 33
± ?? 33
)
Multiplication of Matrix by a Scalar: If we multiply a matrix ?? by a scalar ?? , then each element
of the matrix is multiplied by ??
?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
) ???? = (
????
11
?? ?? 12
?? ?? 13
?? ?? 21
????
22
?? ?? 23
?? ?? 31
?? ?? 32
????
33
)
Multiplication of Two Matrices: Matrix product ???? is possible only if number of columns in
matrix ?? are same as number of rows in matrix ?? .
?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
) ?? = (
?? 11
?? 12
?? 21
?? 22
?? 31
?? 32
)
?? = ???? = (
?? 11
= ?? 11
?? 11
+ ?? 12
?? 21
+ ?? 13
?? 31
?? 12
= ?? 11
?? 12
+ ?? 12
?? 22
+ ?? 13
?? 32
?? 21
= ?? 21
?? 11
+ ?? 22
?? 21
+ ?? 23
?? 31
?? 22
= ?? 21
?? 12
+ ?? 22
?? 22
+ ?? 23
?? 32
)
Note that: (i) ?? ?? ×?? ?? ?? ×?? = ?? ?? ×??
(ii) ???? ? ???? in general
(iii) ???? = 0 does not necessarily imply that ?? = 0 or ?? = 0
(iv) ???? = 0 does not necessarily imply that ???? = 0
For example, ?? = (
0 1
0 0
) ?? = (
1 0
0 0
) ???? = (
0 0
0 0
) ???? = (
0 1
0 0
)
Example 1 If ?? = (
sin?? cos?? sin?? cos?? ) ?? = (
sin?? sin?? cos?? cos ?? ) find ???? and ????
Solution: ???? = (
sin
2
?? + cos
2
?? sin
2
?? + cos
2
??
sin
2
?? + cos
2
?? sin
2
?? + cos
2
?? )= (
1 1
1 1
)
???? = (
2sin
2
?? sin2?? sin2?? 2cos
2
?? )
Example 2 Express the matrix ?? = (
1 3 5
2 -1 4
0 2 3
) as the sum of symmetric and skew-
symmetric matrices.
Solution: ?? = (
1 3 5
2 -1 4
1 2 3
) ?? ?? = (
1 2 1
3 -1 2
5 4 3
)
1
2
( ?? + ?? ?? )= (
1
5
2
3
5
2
-1 3
3 3 3
) ,
1
2
( ?? - ?? ?? )= (
0
1
2
2
-1
2
0 1
-2 -1 0
)
? ?? = (
1
5
2
3
5
2
-1 3
3 3 3
) + (
0
1
2
2
-1
2
0 1
-2 -1 0
)
: Symmetric Skew-Symmetric
4.3 Minors, Cofactors, Determinants and Adjoint of a matrix
Minors associated with elements of a square matrix
A minor of each element of a square matrix is the unique value of the determinant associated with
it, which is obtained after eliminating the row and column in which the element exists.
For a 2 × 2 matrix ?? = (
?? 11
?? 12
?? 21
?? 22
)
?? 11
= ?? 22
, ?? 12
= ?? 21
, ?? 21
= ?? 12
, ?? 22
= ?? 11
For a 3 × 3 matrix ?? = (
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
)
?? 11
= |
?? 22
?? 23
?? 32
?? 33
| , ?? 12
= |
?? 21
?? 23
?? 31
?? 33
| ,…, ?? 33
= |
?? 11
?? 12
?? 21
?? 22
|
Cofactors associated with elements of a square matrix
The cofactor of each element is obtained on multiplying its minor by ( -1)
?? +?? .
?? ????
= ( -1)
?? +?? ?? ????
Determinant of a square matrix
Every square matrix is associated with a determinant and is denoted by det ( ?? ) or |?? |.
det ( ?? )= |?? | = |
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
|
Determinant of order ?? can be expanded by any one row or column using the formula
|?? | = ? ?? ????
?? ?? =1
?? ????
, where ?? ????
is the cofactor corresponding to the element ?? ????
.
A determinant of order 2 is evaluated as:
|?? | = |
?? 11
?? 12
?? 21
?? 22
| = ?? 11
?? 22
- ?? 12
?? 21
A determinant of order 3 is evaluated as:
|?? | = |
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
| = ? ( -1)
?? +?? ?? ????
?? ????
?? ?? =1
=?? 11
|
?? 11
?? 12
?? 21
?? 22
| - ?? 12
|
?? 11
?? 12
?? 21
?? 22
| + ?? 13
|
?? 11
?? 12
?? 21
?? 22
|
=?? 11
( ?? 22
?? 33
- ?? 32
?? 23
)- ?? 12
( ?? 21
?? 33
- ?? 31
?? 23
)+ ?? 13
( ?? 21
?? 32
- ?? 31
?? 22
)
Note: A determinant may be evaluated using any row or column, value remains the same.
Properties of Determinants
? Value of a determinant remains unchanged if rows and columns are interchanged i.e.
|?? | = |?? ?? |
? If any two rows or columns are interchanged, the value of determinant is multiplied
by ( -1)
? The value of determinant remains unchanged if ?? times elements of a row (column) is
added to another row (column).
? If elements in any row (column) in a determinant are multiplied by a scalar ?? , then
value of determinant is multiplied by ?? . Thus, if each element in the determinant is
multiplied by ?? , value of determinant of order ?? multiplies by ?? ?? i.e., |???? | = ?? ?? |?? |
? If ?? and ?? are square matrices of same order, then |???? | = |?? ||?? |
Adjoint of a square matrix
The adjoint of a square matrix ?? of order ?? is the transpose of the matrix of cofactors of each
element. If ?? 11
, ?? 12
, ?? 13
,…, ?? ????
be the cofactors of elements ?? 11
, ?? 12
, ?? 13
,…, ?? ????
of the matrix
?? . Then adjoint of ?? is given by
???? ?? ( ?? )= (
?? 11
?? 12
… ?? 1??
?? 21
?? 22
… ?? 2??
… … … …
?? ?? 1
?? ?? 2
… ?? ????
)
?? = (
?? 11
?? 21
… ?? ?? 1
?? 12
?? 22
… ?? ?? 2
… … … …
?? 1?? ?? 2??
… ?? ????
)
4.4 Inverse of a Matrix
The inverse of a square matrix ?? of order ?? , denoted by ?? -1
is such that
?? ?? -1
= ?? -1
?? = ?? ?? where ?? ?? is an identity matrix of order ?? .
A matrix is invertible if and only if matrix is non-singular i.e., |?? | ? 0. There are many methods
to find inverse of a square matrix.
4.4.1 Inverse of a matrix using adjoint
Working rule to find inverse of a matrix using adjoint:
1. Calculate |?? |
i. If |?? | = 0 , inverse does not exist
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FAQs on Matrices - Engineering Mathematics - Civil Engineering (CE)
| 1. What are matrices used for in mathematics? |  |
Ans. Matrices are used to represent and solve systems of linear equations, perform transformations in geometry, and analyze data in statistics.
| 2. How do you add and subtract matrices? | |
Ans. To add or subtract matrices, you simply add or subtract the corresponding elements in each matrix. The matrices must have the same dimensions for this operation to be possible.
| 3. What is the identity matrix? | |
Ans. The identity matrix is a square matrix in which all elements are zero except for the diagonal elements, which are all one. It acts as the multiplicative identity element in matrix multiplication.
| 4. How do you multiply matrices? | |
Ans. To multiply matrices, you multiply each element of a row in the first matrix by each element of a column in the second matrix, and then sum the products. The resulting matrix will have dimensions equal to the number of rows in the first matrix and the number of columns in the second matrix.
| 5. Can matrices be used to solve real-world problems? | |
Ans. Yes, matrices are frequently used in various fields such as engineering, economics, computer graphics, and physics to model and solve real-world problems involving multiple variables and equations.
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