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MATRICES 

 

Class 10 Notes 

 Introduction: “MATRIX’ is a Latin word for

womb. The knowledge of matrices is

necessary in various branches of

mathematics. Matrices are one of the most

powerful tools in mathematics. The

evolution of concept of matrices is the

result of an attempt to obtain compact and

simple methods of solving system of linear

equations. Matrix notation and operations

are used in electronic spread sheet

programs in PC which can be used in

different areas of business and science.

Matrices are also used in cryptography. 

 Matrix: A rectangular arrangement of set

of elements in the form of horizontal and

vertical lines is called matrix. The

elements can be numbers or variables. 

 

Example:  

Numbers arranged in the form of

horizontal lines are called rows. Numbers

arranged in the form of vertical lines are

called columns. Elements of a matrix are

represented in pair of brackets (or)

parenthesis.

i.e., [ ] (or) ( ). Matrices is the plural of

matrix. 

 Example:  

 

Matrix A contains 2 rows and 3

columns. 

 Order of a matrix: A matrix having ‘m’

rows and ‘n’ columns is said to be of

order m ⋅ n (m cross n (or) m by n). 

If matrix A has ‘m’ rows and ‘n’

columns then m ⋅ n (m cross n (or) m

by n) is the order of matrix, and is

denoted by A m ⋅n. 

Example:  

Number of rows = 2 

Number of columns = 2 

∴ Order of matrix = 2 ⋅ 2. 

It is denoted by A 2 ⋅ 2 

 

Example: The order of the matrix

is: 

Solution: Number of rows = 2. 

Number of columns = 3. 

∴ Order of matrix B = 2 ⋅ 3. 

 

Generally, we use capital letters to denote

matrices. 

 

It is denote by B 2 ⋅ 3 

 

Example: The order of matrix is: 

Solution: Number of rows = 1. 

Number of columns = 4. 

∴ Order of matrix C = 1 ⋅ 4. 

It is denote by C 1 ⋅ 4 

 Example: The order of matrix D = [ 1 ] is 

Solution: Number of rows = 1 = Number of columns. 

∴ Order of matrix D = 1 ⋅ 1. 

It is denote by D 1 ⋅ 1 

A matrix having m rows and n columns having mn (or) nm number of elements.  

Example:  

Order of matrix = 2 ⋅ 3 The element 6 occurs in first row and third column. Therefore it can be represented as

a 13 = 6.  

General form of a matrix: 

In general, m ⋅ n matrix can be represented as 

 

where  

a ij is the element of the matrix in i th row and j th column i.e., is called element

of the matrix. 

 

Example: Construct a 2 ⋅ 3 matrix whose elements are defined by for the representation of elements. 

Solution: Given  

i = 1, 2; j = 1, 2, 3 

 

 

∴ Required matrix =   

Example: Construct a 3 ⋅ 2 matrix whose elements are defined by  

Solution:  

i = 1, 2, 3; j = 1, 2 

 

 

∴ Required matrix   

 

Example: Construct a 2 ⋅ 2 matrix if for the representation of elements. 

Solution: Given  i = 1, 2; j = 1, 2 

 

∴ Required matrix =  Types of matrices:  

i) Row matrix: A matrix having only one row is called row matrix.

Example:

Order of any row matrix is , where n is number of column, n = 2, 3, 4…. 

 ii) Column matrix: A matrix having only one column is called column

matrix.

Example:

Order of any column matrix is , where m is number of column, m = 2, 3, 4….

 iii) Rectangular matrix: A matrix in which number of rows are not equal to

number of columns is called rectangular matrix.

Example:

 iv) Square matrix: A matrix having equal number of rows and columns is

called a square matrix.

 

is a square matrix if m = n. A matrix of order is called a square

matrix of order m.

Example:  

 

If is a square matrix of order n, then the elements a 11 , a 22, …. a nn constitute

principal diagonal. Hence a ij is an element of the diagonal if i = j (or) non-diagonal if i

≠ j. 

Example:  

, a, e, k are the elements of the principal diagonal.  

 

The sum of elements of the diagonal of a square matrix. A is called trace of A is denoted by 

Tr(A). 

Example:   , find the trace of A? 

Solution:  Elements of principal diagonal are 1, 0, 9. 

Tr(A) = 1 + 0 + 9 = 10. 

 v) Diagonal matrix: If each non-diagonal element of a square matrix is

equal to zero, then the matrix is called a diagonal matrix.

If is a diagonal matrix, then . It is denoted as

.

Example:

Example: Determine the diagonal elements of the matrix 

 

 

Solution: i = j  

a11 = a, a22 = B, a 33 = C1  

∴ Diagonal elements of the matrix are a, B, C 1 . 

 Example: Determine the principal diagonal of the matrix given by  

of order 3 ⋅ 3 

Solution: Given  

 

 

 

 

 vi) Scalar matrix: If each non-diagonal element of a square matrix is zero

and all diagonal elements are same, then it is called a scalar matrix.

A matrix is said to be a scalar matrix if.

 

Example:

vii) Unit matrix (or) Identity matrix: If each non-diagonal element of a

square matrix is zero and each diagonal element is equal to 1, then that matrix is

called a unit or identity matrix. It is denoted by I n (or) I.

If

Example:

Example:  

viii) Zero (or) Null matrix: If each element of a matrix is zero, then it is called

a null matrix or zero matrix. It is denoted by O m ⋅n (or) O.

Example:  

A square matrix is said to be upper triangular if a ij = 0 for all i > j .

A square matrix is said to be lower triangular if aij = 0 for all i < j . Example: 

i) upper triangular matrices. 

 

ii) lower triangular matrices. 

  Comparison of matrices: Two matrices can be compared if the order of matrices is

equal. i.e., they have same number of rows and same number of columns. 

Example:  

Compared because order of matrices is equal. 

  Equality of matrices: Matrices A and B are said to be equal if A and B are of same

order and the corresponding elements of A and B are equal. 

Thus, A = B if aij = bij .  

Example:  

 

 

Example: If such that A = B then find x, y, z . 

Solution:  

Given A = B, corresponding elements are equal. 

 

 

 

 

Multiplication of a matrix by a scalar:

If every element of a matrix A is multiplied by a number (real or complex) k, the

matrix obtained is k times A and is denoted by kA and the operation is called scalar

multiplication. 

Example:  

1.

Solution: 

 

 

 

   

1. If a and b are any two scalars and P is a matrix, then a(bP) = (ab)P

2. If m and n are any two scalars and A is a matrix, then (m + n)A = mA +

nA.

  

 

Addition of matrices: Two matrices can be added if the order of matrices are equal. 

 

The sum matrix of two matrices A and B is obtained by adding the corresponding

elements of A and B. 

If A = and B = then 

A + B can be represented as such that i = 1, 2, 3, ….. m, j =1, 2, 3, ….. n. 

Example:  

 

Example: Find the sum matrix of matrices and  

Solution: The given two matrices are of same order. So they can be added. 

Sum =  

 Example: Find the sum of matrices A = aij = 2i + j and B = bij =   j

i   .

Solution: A = [ a ij ] 

Given a ij = 2i + j  

i = 1, 2; j = 1, 2 

 

B = [ bij ]  

Given aij =   ji  

 

i = 1, 2; j = 1, 2 

 

 

 

  

Properties on addition of matrices:  

i) Closure property: When we add any two matrices of same order then the

resultant also is matrix of same order.

Example:

 ii) Commutative property: Let A, B be two matrices then A + B = B + A

Example:

 iii) Associative property: Let A, B, C be three matrices then (A + B) + C = A +

(B + C)

 

Example:

   

iv) Additive identity: Let A be a matrix and O is zero matrix then A + O = O + A = A

Example:

 

v) Additive inverse: Let A be a matrix then A + (-A) = (-A) + A = O

∴ (-A) is called additive inverse of A.

A is called additive inverse of –A.

Example:  

 

 

 

 vi) Cancellation laws:

A + B = A + C ⇒ B = C (Left cancellation law)

B + A = C + A ⇒ B = C (Right cancellation law)

  

Subtraction of matrices: Two matrices can be subtracted if they are of the same order. 

The difference of two matrices of same order A and B i.e. A – B is obtained by

subtracting the corresponding elements of B from that of A. Also the difference matrix

is of the same order as that of A or B. 

If then A – B can be represented as such that  

i = 1, 2, 3, ……m; j =1, 2, 3, ……n. 

Example:  

 

Example: Find the subtraction of matrices  

Solution:  

 

 

 i)  

Matrix subtraction is not commutative. 

ii)  

Matrix subtraction is not associative.  Example: Find the subtraction of matrices if  

 

Solution: Given  

 

 

 

 

 

 

 

 

 

  Aliter: 

 

Example: Find A + (-B) if  

Solution: Given  

 

 

 

 A + (-B) = A - B  Solving a matrix equation: Let A, B be two matrices of same order such that A + X = B

where X is an unknown matrix or order equal to A and B matrices. 

 

Example: find X of order 2 ⋅ 2 such that  

i) X = A – B ii) A + X = B 

 

Solution: Given  

i) X = A – B  

 

  

ii) A + X = B 

 

  Transpose of a matrix: The matrix obtained from any given matrix A by interchanging

its rows and columns is called transpose of the given matrix. It is denoted by A T .

If then A T =  

Example:

Solution:  

Example: If then find matrix. 

Solution:  

  Properties of transpose matrix:  

i) If A is any matrix then

ii) If A, B are two matrices of same order then

 

iii) If A, B are two matrices of same order then

iv) If A is a matrix and k is a scalar then

Example: If then find  

i) ii) iii)  

Solution: i) Given  

 

ii)  

 

iii)  

 

Example: If and k is a scalar then verify that  

Solution: Given  

 

 

 Symmetric matrix: A square matrix is said to be symmetrix if the transpose of the

matrix is equal to itself. 

A square matrix is said to be symmetric if its ( i, j) th element is same as its

(j, i) th element, i.e., aij = aji , for all i, j.  

Example:  

 

∴ A is symmetric matrix. 

 

 

i) Symmetric matrix is always a square matrix.

ii) A necessary and sufficient condition for a matrix A to be symmetric is

that it is equal to its transpose matrix. i.e., .

iii) Diagonal matrices are always symmetric.

  Skew-symmetric matrix: If a square matrix and the negative of its transpose matrix

are equal then it is called a skew-symmetric matrix. 

A square matrix A = [ aij ] is said to be skew-symmetric if the (i, j) th

element of A is the negative of the (j, i) th element of A. i.e., aij = -aji , for all i, j .  

 

Example:  

 

∴ A is skew symmetric matrix. 

 A matrix which is both symmetric and skew symmetric is called a square null matrix. 

 

 

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