Flashcards: Matrices | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


 
 
 
 
MATRICES 
A set of ???? numbers (real or complex) arranged in the form of a rectangular array having ' ?? ' rows and ' 
?? ' columns is called an ?? × ?? matrix. It is usually written as 
?? = [
?? 11
?? 12
? ?? 1?? ?? 21
?? 22
? ?? 2?? ? ? ? ?
?? ?? 1
?? ?? 2
? ?? ????
] 
  
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MATRICES 
A set of ???? numbers (real or complex) arranged in the form of a rectangular array having ' ?? ' rows and ' 
?? ' columns is called an ?? × ?? matrix. It is usually written as 
?? = [
?? 11
?? 12
? ?? 1?? ?? 21
?? 22
? ?? 2?? ? ? ? ?
?? ?? 1
?? ?? 2
? ?? ????
] 
  
 
 
 
 
MATRICES 
It is represented as ?? = [?? ????
]
?? ×?? . The numbers ?? 11
,?? 12
,… etc. are called elements of the matrix. The 
element ?? ????
 belong to ?? th 
 row and ?? th 
 column. 
e.g., ?? = [
3 2 7
5 -4 6
4 8 -12
] is a 3 × 3 matrix 
  
Page 3


 
 
 
 
MATRICES 
A set of ???? numbers (real or complex) arranged in the form of a rectangular array having ' ?? ' rows and ' 
?? ' columns is called an ?? × ?? matrix. It is usually written as 
?? = [
?? 11
?? 12
? ?? 1?? ?? 21
?? 22
? ?? 2?? ? ? ? ?
?? ?? 1
?? ?? 2
? ?? ????
] 
  
 
 
 
 
MATRICES 
It is represented as ?? = [?? ????
]
?? ×?? . The numbers ?? 11
,?? 12
,… etc. are called elements of the matrix. The 
element ?? ????
 belong to ?? th 
 row and ?? th 
 column. 
e.g., ?? = [
3 2 7
5 -4 6
4 8 -12
] is a 3 × 3 matrix 
  
 
 
 
TYPES OF MATRICES 
1 Square Matrix : A matrix having equal number of rows and columns is called a square matrix. 
If the matrix ?? has ?? rows and ?? columns, it is said to be square matrix of order ?? . 
Example : [
1 2 3
2 3 4
3 4 5
] is a square matrix of order 3. 
The diagonal of this matrix containing the elements 1,3,5 is called the leading or principal diagonal. 
Trace of matrix : The sum of the diagonal elements of a square matrix ?? is called trace of ?? . 
Trace? (?? ) = ?
?? =1
?? ??? ????
 
  
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MATRICES 
A set of ???? numbers (real or complex) arranged in the form of a rectangular array having ' ?? ' rows and ' 
?? ' columns is called an ?? × ?? matrix. It is usually written as 
?? = [
?? 11
?? 12
? ?? 1?? ?? 21
?? 22
? ?? 2?? ? ? ? ?
?? ?? 1
?? ?? 2
? ?? ????
] 
  
 
 
 
 
MATRICES 
It is represented as ?? = [?? ????
]
?? ×?? . The numbers ?? 11
,?? 12
,… etc. are called elements of the matrix. The 
element ?? ????
 belong to ?? th 
 row and ?? th 
 column. 
e.g., ?? = [
3 2 7
5 -4 6
4 8 -12
] is a 3 × 3 matrix 
  
 
 
 
TYPES OF MATRICES 
1 Square Matrix : A matrix having equal number of rows and columns is called a square matrix. 
If the matrix ?? has ?? rows and ?? columns, it is said to be square matrix of order ?? . 
Example : [
1 2 3
2 3 4
3 4 5
] is a square matrix of order 3. 
The diagonal of this matrix containing the elements 1,3,5 is called the leading or principal diagonal. 
Trace of matrix : The sum of the diagonal elements of a square matrix ?? is called trace of ?? . 
Trace? (?? ) = ?
?? =1
?? ??? ????
 
  
 
 
 
 
2 Diagonal Matrix : A square matrix in which all the elements, except those in the leading 
diagonal are zero is called a diagonal matrix. Thus square matrix is called diagonal matrix if ?? ????
= 0 
for ?? ? ?? . 
e.g., ?? = [
2 0 0
0 1 0
0 0 4
] or [
?? 1
0 0
0 ?? 2
0
0 0 ?? 3
] 
Above are diagonal matrices of the type 3 × 3. These are in short written as diag? [2,1,4] or diag 
[?? 1
,?? 2
,?? 3
] 
If the diagonal elements are ?? 1
,?? 2
,…?? ?? , then it is written as diag? [?? 1
,?? 2
,… ,?? ?? ]. 
 
 
 
 
 
Page 5


 
 
 
 
MATRICES 
A set of ???? numbers (real or complex) arranged in the form of a rectangular array having ' ?? ' rows and ' 
?? ' columns is called an ?? × ?? matrix. It is usually written as 
?? = [
?? 11
?? 12
? ?? 1?? ?? 21
?? 22
? ?? 2?? ? ? ? ?
?? ?? 1
?? ?? 2
? ?? ????
] 
  
 
 
 
 
MATRICES 
It is represented as ?? = [?? ????
]
?? ×?? . The numbers ?? 11
,?? 12
,… etc. are called elements of the matrix. The 
element ?? ????
 belong to ?? th 
 row and ?? th 
 column. 
e.g., ?? = [
3 2 7
5 -4 6
4 8 -12
] is a 3 × 3 matrix 
  
 
 
 
TYPES OF MATRICES 
1 Square Matrix : A matrix having equal number of rows and columns is called a square matrix. 
If the matrix ?? has ?? rows and ?? columns, it is said to be square matrix of order ?? . 
Example : [
1 2 3
2 3 4
3 4 5
] is a square matrix of order 3. 
The diagonal of this matrix containing the elements 1,3,5 is called the leading or principal diagonal. 
Trace of matrix : The sum of the diagonal elements of a square matrix ?? is called trace of ?? . 
Trace? (?? ) = ?
?? =1
?? ??? ????
 
  
 
 
 
 
2 Diagonal Matrix : A square matrix in which all the elements, except those in the leading 
diagonal are zero is called a diagonal matrix. Thus square matrix is called diagonal matrix if ?? ????
= 0 
for ?? ? ?? . 
e.g., ?? = [
2 0 0
0 1 0
0 0 4
] or [
?? 1
0 0
0 ?? 2
0
0 0 ?? 3
] 
Above are diagonal matrices of the type 3 × 3. These are in short written as diag? [2,1,4] or diag 
[?? 1
,?? 2
,?? 3
] 
If the diagonal elements are ?? 1
,?? 2
,…?? ?? , then it is written as diag? [?? 1
,?? 2
,… ,?? ?? ]. 
 
 
 
 
 
 
 
 
 
 
3 Scalar Matrix : A diagonal matrix in which all the elements in the leading diagonal are equal is 
called scalar matrix. It is equal to ?? ?? ?? for some scalar ?? . 
e.g., ?? = [
3 0 0
0 3 0
0 0 3
] or [
?? 0 0
0 ?? 0
0 0 ?? ] 
In general for a scalar matrix, ?? ????
= 0 for ?? ? ?? and ?? ????
= ?? for ?? = ?? .