ICSE Revision Notes for Matrices Class 10 Maths
Chapter Name  Matrices 
Topics Covered 

Related Study 
Matrix and its Various Types
Matrices and their Related Terminology
A matrix is an ordered rectangular array of numbers and functions. The numbers or functions in a matrix are called the elements of the matrix.
For example, if the marks obtained by Smita in English, Maths and Science are 84, 93, and 89 respectively and the marks scored by Gopal are 81, 90, and 92 respectively, then this can be represented in the form of matrix asor as
Here, in the first matrix, the vertical lines of elements represent the marks obtained by Smita and Gopal respectively. In the second matrix, the horizontal lines of elements represent the marks obtained by Smita and Gopal.
The matrices are usually denoted by capital letters. The horizontal line of elements is known as the row of matrix and the vertical line of elements is known as the column of matrix.
A matrix with m rows and n columns is known as the matrix of order m × n or an m × n matrix.
For example: is a 3 × 2 matrix and is a 2 × 3 matrix.
In general, an m × n matrix represents the matrix
It can also be represented as A = [a_{ij}]_{m}_{×}_{n}, 1 ≤ i ≤ m, 1 ≤ j ≤ n; i, j ∈ N. .
This means that the i^{th }row has elements a_{i}_{1}, a_{i}_{2} … a_{ij} … a_{in} and the j^{th }column has elements a_{1j}, a_{2j}… a_{ij} … a_{mj}.
An m × n matrix has mn number of elements.
Types of Matrices
A matrix is said to be a column matrix if it has only one column.
For example:
For example:
A matrix is said to be a rectangular matrix if the number of rows is not equal to the number of columns.
For example:
In a square matrix A = [a_{ij}]_{n×n} , the elements a_{11}, a_{22} … a_{nn} are called the diagonal elements of A = [a _{ij}]_{n×n}. For instance, for the square matrix , the diagonal elements are 3, 4, and 8. If all the nondiagonal elements of a square matrix are zero, then the square matrix is known as a diagonal matrix i.e., a square matrix A = [a_{ij }]_{n×n} is a diagonal matrix if a_{ij }= 0, whenever i ≠ j.
For example:
A diagonal matrix in which all the diagonal elements are equal is known as a scalar matrix i.e., square matrix A = [a_{ij }]_{n×n }is a scalar matrix if
a_{ij} = 0, when i ≠ j
a_{ij} = k, when i = j, where k is a constant.
For example:
A square matrix in which all the diagonal elements are 1 and all other elements are zero is known as an identity matrix i.e., A = [a_{ij}]_{n×n} is an identity matrix if
a_{ij} = 1, when i = j
a_{ij} = 0, when i ≠ j
For example:
A matrix is known as a zero or null matrix if all its elements are zero.
For example:, are all zero matrices.
Let us solve some examples on matrices.
Example 1: The lengths of the sides of two triangles are 2 cm, 5 cm, 8 cm and 3 cm, 4 cm, 5 cm. Represent this in the form of a 2 × 3 matrix.
Answer
A 2 × 3 matrix is represented as
Example 2: Construct a square matrix of order 3 whose elements are given by a_{ij} = 1/2 2i – j.
Answer
A 3 × 3 matrix is represented as
a_{11} = 1/22.1 – 1 = ½, a_{12} = 1/22.1 – 2 = 0, a_{13} = ½2.1 – 3 = ½ 1 = 1/2
a_{21} = 1/22.2 – 1 = 3/2, a_{22} = ½2.2 – 2 = 1, a_{23} = 1/22.2 – 3= 1/2
a_{31} = 1/22.3 – 1 = 5/2. a_{32} = 1/22.3 2 = 2, a_{31} = 1/22.3 – 3 = 3/2
Thus, the required matrix is
Transpose of a Matrix and Related Properties
Transpose of a Matrix
If A = [a_{ij}]_{m}_{×}_{n }is a matrix of order m × n, then the matrix obtained by interchanging its rows and columns is known as the transpose of A.
It is denoted by A' . In general, if A = [a_{ij}]_{m}_{×}_{n}, then A' = [a_{ij}]_{m}_{×}_{n}
For example:
Note: Transpose of a row matrix is a column matrix and vice versa.
For example,
Solved Examples
Example 1:
If, then prove that (A + B)' = A' + B'
Answer
From (1) and (2), we obtain(A + B)' = A' + B'
Thus, given result is proved.
Example 2: If then prove that (AB)' = B'A'
The transpose of A is given by
Example 4: Let m × n be the order a matrix A. If the order of the transpose of the matrix A is 4 × 3 then find the value of m and n.
Answer
Order of the matrix A = m × n
Order of transpose of the matrix A = n × m = 4 × 3
Therefore, n = 4 and m = 3.
Example 5: Let A =, then show that (A')' + A =
Answer
Here,
Equality of Matrices
Two matrices A = [a_{ij}] and B = [b_{ij}] and are said to be equal (denoted as A = B) if they are of the sameorder and each element of A is equal to the corresponding element of B i.e., a_{ij }= b_{ij} for all i and j.
For example:
Solved Examples
Example 1: If, then find the values of x, y and z.
Answer
Since the corresponding elements of equal matrices are equal,
2x + y = 7…(1)
x − y = 5…(2)
2x + y + z = 13…(3)
3y + z = 3…(4)
On solving equations (1) and (2), we obtain x = 4 and y = −1.
On substituting the value of y in equation (4), we obtain z = 6.
Thus, the values of x, y and z are 4, −1 and 6 respectively.
Example 2: Ifare two equal matrices, then find matrix A and matrix B.
Answer
The given matrices are equal. Therefore, their corresponding elements are equal. On comparing the corresponding elements, we obtain
a + b = −a
⇒ 2a + b = 0…(1)
2b − c = 3c
⇒ 4c − 2b = 0…(2)
b = −b + a + 5
⇒ −a + 2b = 5…(3)
On solving equations (1) and (3), we obtain a = −1 and b = 2.
On substituting the value of b in equation (2), we obtain c = 1.
In matrix A,
a_{11} = a + b = −1 + 2 = 1
a_{12} = 2b − c = 2(2) − 1 = 3
a_{22} = b = 2
Since A and B are equal matrices, B is the same as matrix A.
Thus, matrix A and matrix B are
Addition and Subtraction of Matrices
Addition of Matrices and its Properties
The sum of two matrices is obtained by adding their corresponding elements. Two matrices can be added only if their orders are same.
In general, if A = [a_{ij}] and B = [b_{ij}] and are two matrices of order m × n, then the sum of these matrices is given by C = [c _{ij}]_{m×n}, where c_{ij} = a_{ij} + b_{ij}.
For example:
The sum of matrices whose order is not the same is not defined.
If A = [a_{ij}], B = [b_{ij}], C = [c_{ij}] are three matrices of the same order, then their addition satisfies the following properties:
The addition of matrices is commutative i.e., A + B = B + A
The addition of matrices is associative i.e., (A + B) + C = A + (B + C)
For every matrix A, there exists a zero matrix O of the same order such that A + O = O + A = A. O is the additive identity of the matrix addition.
For every matrix A, there exists another matrix −A of same order such that A + (−A) = (−A) + A = O. (−A) is the additive inverse of A.
Subtraction of Matrices
The difference of two matrices A = [a_{ij}] and B = [b_{ij}] of the same order m × n, (say B from A) is given by, where c_{ij }= a_{ij }− b_{ij}.
For example:
The difference of matrices whose orders are not the same is not defined.
• A − B ≠ B − A
Solved Examples
Example 1:
If , then find the values of x and y.
Answer
According to the given information,
On equating the corresponding elements, we obtain
x – 4y =  3 ...(1)
2x + 3y = 5 ....(2)
On equating (1) and (2), we obtain x = 1 and y = 1
Thus, the values of x and y are 1 and 1 respectively.
Example 2:
If are three matrices, then prove that A + B  C is an identity matrix.
Answer
Multiplication of a Matrix by a Scalar and Related Properties
Multiplication of a Matrix by a Scalar
A matrix A, when multiplied by a scalar k, gives another matrix kA, which is obtained by multiplying each element of matrix A by the scalar k. In general, if A = [a_{ij}]_{m×n}, then kA = k[a_{ij}]_{m×n} = [k(a_{ij})]_{m×n }i.e., (i, j)^{th} , element of kA is ka_{ij}, for all values of i and j.
For example:
The negative of a matrix is obtained by multiplying the matrix with −1 i.e., if A is a matrix, then its negative is (−1)A = −A.
If A = [a_{ij}] and B = [b_{ij}] are two matrices of the same order i.e., m × n, and k and l are scalars, then A and B satisfies the following properties:
k (A + B) = kA + kB
(k + l) A = kA + lA
Solved Examples
Example 1:
Find the value of 2A − 3B, where
Answer
If , then find the values of a and b.
Answer
On equating the corresponding elements, we obtain−6 − 5a = −1
⇒ a = −1
9 − 5b = −1
⇒ b = 2
Thus, the values of a and b are −1 and 2 respectively.
Multiplication of Matrices and Related Properties
Multiplication of Matrices
The product of two matrices A and B is defined only if the number of columns of A is equal to the number of rows of B i.e., if A = [a_{ij} ]_{m}_{×n} and B = [b_{jk}]_{n}_{×p}, only then AB is defined.
If A is a matrix of order m×n and B is a matrix of order n×p, then their product, say, C = AB, is of the order m×p. To find the (i, k)^{th }element, (c_{ik}) of matrix C (= AB), multiply the elements of the i^{th }row of A and the elements of the k^{th }column of B and take the sum of all their products.
If the product of two matrices is a zero matrix, then it is not necessary that one of the matrices is a zero matrix.
Properties of Multiplication of Matrices
The multiplication of three matrices A, B and C satisfies the following properties:
The multiplication of matrices is noncommutative i.e., the product AB and BA may or may not be equal.
For example:
The multiplication of diagonal matrices of the same order is commutative.∴ 4x + 3y = 7 ...(1)
− 2x + y = 9 ...(2)
4y = 20 ...(3)
From equation (3), we obtain y = 5.
On substituting this value in equation (2), we obtain x = −2.
Thus, the values of x and y are −2 and 5 respectively.
Example 2:
If A = , then prove that A(B + C) = AB + BC.
Answer
From (1) and (2), we obtain
A(B + C) = AB + AC
Thus, the result is proved.