Matrices: Chapter 3 Links
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This post provides step-by-step NCERT Solutions for Class 12 Maths Matrices 3.1. It will help you understand important matrix concepts and solve similar questions with confidence.
Key Concepts
- 1. Matrix Notation
A matrix is a rectangular arrangement of numbers, symbols or expressions in rows or columns enclosed within brackets.
A matrix is generally denoted by capital letters like A,\ B,\ C etc.
For example,
A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}
- 2. Order of a Matrix
The order of a matrix is written as:
\text{Number of rows}\times\text{Number of columns}
If a matrix has m rows and n columns, then its order is m\times n .
For example,
\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} is of order 2\times3 .
- 3. Row Matrix
A matrix having only one row is called a row matrix.
For example,
\begin{bmatrix} 2 & 5 & -1 & 7 \end{bmatrix}
- 4. Column Matrix
A matrix having only one column is called a column matrix.
For example,
\begin{bmatrix} 3 \\ -1 \\ 4 \end{bmatrix}
- 5. Square Matrix
A matrix having equal number of rows and columns is called a square matrix.
For example,
\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
💡Identity Matrix ⊂ Scalar Matrix ⊂ Diagonal Matrix ⊂ Square Matrix
- 6. Diagonal Matrix
A square matrix in which all non-diagonal elements are zero is called a diagonal matrix.
For example,
\begin{bmatrix} 2 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & -1 \end{bmatrix}
- 7. Scalar Matrix
A diagonal matrix in which all diagonal elements are equal is called a scalar matrix.
For example,
\begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix}
- 8. Identity Matrix
A square matrix in which all diagonal elements are 1 and all other elements are 0 is called an identity matrix.
It is usually denoted by I .
For example,
I= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
- 9. Zero Matrix
A matrix in which all elements are zero is called a zero matrix or null matrix.
For example,
\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
- 10. Equality of Matrices
Two matrices are said to be equal if:
- They are of the same order
- Their corresponding elements are equal
If
A=[a_{ij}] \text{ and } B=[b_{ij}]
then
A=B \iff a_{ij}=b_{ij} \text{ for all } i,j
Now that we have revised the concepts, let’s begin Matrices 3.1 Exercise with Question 1.
Question 1: Matrices 3.1
1. In the matrix
A=\begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}
Write:
- (i) The order of the matrix
- (ii) The number of elements
- (iii) The elements a_{13}, a_{21}, a_{33}, a_{24}, a_{23}
Solution:
(i) Order of the matrix
We observe that the given matrix has 3 rows and 4 columns.
\therefore The order of the matrix is 3 \times 4 .
(ii) Number of elements
We know that:
\text{Number of elements} = \text{Number of rows} \times \text{Number of columns}
Here, number of rows =3 and number of columns =4 .
\Rightarrow 3 \times 4 = 12
\therefore The matrix has 12 elements.
(iii) Required elements
We know that a_{ij} represents the element in the i^{th} row and j^{th} column. Therefore,
- a_{13}=19
- a_{21}=35
- a_{33}=-5
- a_{24}=12
- a_{23}=\frac{5}{2}
Question 2: Order of Matrices
2. If a matrix has 24 elements, what are the possible orders it can have? What if it has 13 elements?
Solution:
(i) Matrix having 24 elements
Let the order of the matrix be m \times n .
We know that:
m \times n = 24
The possible factor pairs of 24 are:
(1,24), (2,12), (3,8), (4,6) and their reverses.
\therefore Possible orders are:
- 1 \times 24
- 24 \times 1
- 2 \times 12
- 12 \times 2
- 3 \times 8
- 8 \times 3
- 4 \times 6
- 6 \times 4
(ii) Matrix having 13 elements
We know that:
m \times n = 13
Since 13 is a prime number, its only factors are 1 and 13.
\therefore Possible orders are:
- 1 \times 13
- 13 \times 1
You may already be following Maths Better for important concepts and exam-oriented questions, ranging from integration tricks to solving linear equations using the Matrix Method. Similarly, this NCERT Solutions series for Class 12 Maths will help you strengthen both concepts and problem-solving skills.
Now, let’s move on to the next question of Matrices 3.1.
Question 3: Elements and Order of Matrices
3. If a matrix has 18 elements, what are the possible orders it can have? What if it has 5 elements?
Solution:
(i) Matrix having 18 elements
Let the order of the matrix be m \times n .
We know that:
m \times n = 18
The possible factor pairs of 18 are:
(1,18), (2,9), (3,6) and their reverses.
\therefore Possible orders are:
- 1 \times 18
- 18 \times 1
- 2 \times 9
- 9 \times 2
- 3 \times 6
- 6 \times 3
(ii) Matrix having 5 elements
We know that:
m \times n = 5
Since 5 is a prime number, its only factors are 1 and 5.
\therefore Possible orders are:
- 1 \times 5
- 5 \times 1
Question 4: Matrices 3.1
4. Construct a 2 \times 2 matrix A=[a_{ij}] , whose elements are given by:
- (i) a_{ij}=\frac{(i+j)^2}{2}
- (ii) a_{ij}=\frac{i}{j}
- (iii) a_{ij}=\frac{(i+2j)^2}{2}
Solution:
(i) a_{ij}=\frac{(i+j)^2}{2}
Since the matrix is of order 2 \times 2 , we take:
i,j=1,2
Now,
- a_{11}=\frac{(1+1)^2}{2}=\frac{4}{2}=2
- a_{12}=\frac{(1+2)^2}{2}=\frac{9}{2}
- a_{21}=\frac{(2+1)^2}{2}=\frac{9}{2}
- a_{22}=\frac{(2+2)^2}{2}=\frac{16}{2}=8
\therefore The required matrix is:
A=\begin{bmatrix} 2 & \frac{9}{2} \\ \frac{9}{2} & 8 \end{bmatrix}
(ii) a_{ij}=\frac{i}{j}
Now,
- a_{11}=\frac{1}{1}=1
- a_{12}=\frac{1}{2}
- a_{21}=\frac{2}{1}=2
- a_{22}=\frac{2}{2}=1
\therefore The required matrix is:
A=\begin{bmatrix} 1 & \frac{1}{2} \\ 2 & 1 \end{bmatrix}
(iii) a_{ij}=\frac{(i+2j)^2}{2}
Now,
- a_{11}=\frac{(1+2\times1)^2}{2}=\frac{9}{2}
- a_{12}=\frac{(1+2\times2)^2}{2}=\frac{25}{2}
- a_{21}=\frac{(2+2\times1)^2}{2}=\frac{16}{2}=8
- a_{22}=\frac{(2+2\times2)^2}{2}=\frac{36}{2}=18
\therefore The required matrix is:
A=\begin{bmatrix} \frac{9}{2} & \frac{25}{2} \\ 8 & 18 \end{bmatrix}
Question 5: Construction of Matrices
5. Construct a 3 \times 4 matrix, whose elements are given by:
- (i) a_{ij}=\frac{1}{2}|-3i+j|
- (ii) a_{ij}=2i-j
Solution:
(i) a_{ij}=\frac{1}{2}|-3i+j|
Since the matrix is of order 3 \times 4 , we take:
i=1,2,3 \quad \text{and} \quad j=1,2,3,4
Now,
- a_{11}=\frac{1}{2}|-3(1)+1|=\frac{1}{2}|-2|=1
- a_{12}=\frac{1}{2}|-3(1)+2|=\frac{1}{2}|-1|=\frac{1}{2}
- a_{13}=\frac{1}{2}|-3(1)+3|=\frac{1}{2}(0)=0
- a_{14}=\frac{1}{2}|-3(1)+4|=\frac{1}{2}(1)=\frac{1}{2}
- a_{21}=\frac{1}{2}|-3(2)+1|=\frac{1}{2}|-5|=\frac{5}{2}
- a_{22}=\frac{1}{2}|-3(2)+2|=\frac{1}{2}|-4|=2
- a_{23}=\frac{1}{2}|-3(2)+3|=\frac{1}{2}|-3|=\frac{3}{2}
- a_{24}=\frac{1}{2}|-3(2)+4|=\frac{1}{2}|-2|=1
- a_{31}=\frac{1}{2}|-3(3)+1|=\frac{1}{2}|-8|=4
- a_{32}=\frac{1}{2}|-3(3)+2|=\frac{1}{2}|-7|=\frac{7}{2}
- a_{33}=\frac{1}{2}|-3(3)+3|=\frac{1}{2}|-6|=3
- a_{34}=\frac{1}{2}|-3(3)+4|=\frac{1}{2}|-5|=\frac{5}{2}
\therefore The required matrix is:
A=\begin{bmatrix} 1 & \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{5}{2} & 2 & \frac{3}{2} & 1 \\ 4 & \frac{7}{2} & 3 & \frac{5}{2} \end{bmatrix}
(ii) a_{ij}=2i-j
Now,
- a_{11}=2(1)-1=1
- a_{12}=2(1)-2=0
- a_{13}=2(1)-3=-1
- a_{14}=2(1)-4=-2
- a_{21}=2(2)-1=3
- a_{22}=2(2)-2=2
- a_{23}=2(2)-3=1
- a_{24}=2(2)-4=0
- a_{31}=2(3)-1=5
- a_{32}=2(3)-2=4
- a_{33}=2(3)-3=3
- a_{34}=2(3)-4=2
\therefore The required matrix is:
A=\begin{bmatrix} 1 & 0 & -1 & -2 \\ 3 & 2 & 1 & 0 \\ 5 & 4 & 3 & 2 \end{bmatrix}
The questions in Matrices 3.1 are simple and scoring once the concepts are clear. However, Mathematics becomes easier only with practice. So, solve these NCERT questions regularly and try to understand each step carefully. If you have any doubt, feel free to leave a comment, I’ll be happy to help.
Question 6: Matrices 3.1
6. Find the values of x, y and z from the following equations:
- (i) \begin{bmatrix} 4 & 3 \\ x & 5 \end{bmatrix} = \begin{bmatrix} y & z \\ 1 & 5 \end{bmatrix}
- (ii) \begin{bmatrix} x+y & 2 \\ 5+z & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}
- (iii) \begin{bmatrix} x+y+z \\ x+z \\ y+z \end{bmatrix} = \begin{bmatrix} 9 \\ 5 \\ 7 \end{bmatrix}
Solution:
(i) We know that equal matrices have equal corresponding elements.
- Comparing first row first column:
y=4
- Comparing first row second column:
We get, z=3
- Comparing second row first column:
x=1
\therefore x=1,\ y=4,\ z=3
(ii) Comparing corresponding elements, we get:
x+y=6 \quad ...(1)
5+z=5
That \Rightarrow z=0
Also,
xy=8 \quad ...(2)
From equations (1) and (2), the numbers whose sum is 6 and product is 8 are 2 and 4.
\therefore x=2,\ y=4
(or x=4,\ y=2 )
Also, z=0 .
(iii) Comparing corresponding elements, we get:
x+y+z=9 \quad ...(1)
x+z=5 \quad ...(2)
and y+z=7 \quad ...(3)
Subtracting equation (2) from equation (1), we get:
y=9-5=4
Substituting y=4 in equation (3),
We get, 4+z=7
\Rightarrow z=3
Substituting z=3 in equation (2),
gives x+3=5
\Rightarrow x=2
\therefore x=2,\ y=4,\ z=3
Question 7: Matrices 3.1
7. Find the values of a, b, c and d from the equation:
\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}
Solution
We know that equal matrices have equal corresponding elements.
Comparing corresponding elements, we get:
a-b=-1 \quad ...(1)
2a+c=5 \quad ...(2)
And 2a-b=0 \quad ...(3)
Also 3c+d=13 \quad ...(4)
Subtracting equation (1) from equation (3), we get:
(2a-b)-(a-b)=0-(-1)
\Rightarrow a=1
Substituting a=1 in equation (1),
1-b=-1
\Rightarrow b=2
Substituting a=1 in equation (2),
We get, 2(1)+c=5
\Rightarrow 2+c=5
\Rightarrow c=3
Substituting c=3 in equation (4),
We get, 3(3)+d=13
i.e. 9+d=13
\Rightarrow d=4
\therefore a=1,\ b=2,\ c=3,\ d=4
Question 8: Ex 3.1 Matrices – MCQ
8. A=[a_{ij}]_{m\times n} is a square matrix, if
- (A) m<n
- (B) m>n
- (C) m=n
- (D) None of these
Answer:
✅ Correct option: (C) m=n
Explanation:
A matrix is called a square matrix when the number of rows is equal to the number of columns.
\therefore m=n
NCERT Class 12 Maths has a total of 61 exercises across Part 1 and Part 2. I’ll be covering each exercise one by one with clear explanations and exam-oriented solutions. Many of these questions are also available in video format on my YouTube channel, @Mathsbetter.
Here’s the next question.
Question 9: Matrices 3.1 – MCQ
9. Which of the given values of x and y make the following pair of matrices equal?
\begin{bmatrix} 3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix} = \begin{bmatrix} 0 & y-2 \\ 8 & 4 \end{bmatrix}
- (A) x=-\frac{1}{3},\ y=7
- (B) Not possible to find
- (C) y=7,\ x=-\frac{2}{3}
- (D) x=-\frac{1}{3},\ y=-\frac{2}{3}
Answer:
✅ Correct option: (B) Not possible to find
Explanation:
For two matrices to be equal, each corresponding element must be equal.
From first row first column,
3x+7=0
i.e. 3x=-7
\Rightarrow x=-\frac{7}{3}
From second row second column,
2-3x=4
So, -3x=2
\Rightarrow x=-\frac{2}{3}
The two values of x are different.
\therefore No common solution exists.
The following is the last question of Matrices 3.1, which uses the Fundamental Principle of Counting (FPC).
Question 10: Matrices 3.1 – MCQ
10. The number of all possible matrices of order 3\times3 with each entry 0 or 1 is:
- (A) 27
- (B) 18
- (C) 81
- (D) 512
Answer:
✅ Correct option: (D) 512
Explanation:
A matrix of order 3\times3 has:
3\times3=9 entries.
Each entry can be filled in 2 ways:
- 0
- 1
Therefore, total possible matrices (by Fundamental Principle of Counting)
=2^9=512
Common Mistakes to Avoid
- Confusing rows and columns
Always count horizontal entries as rows and vertical entries as columns while finding the order of a matrix. - Writing incorrect order of a matrix
The order is written as:
\text{Number of rows}\times\text{Number of columns}
and not vice versa. - Confusing row matrix and column matrix
A row matrix has only one row, whereas a column matrix has only one column. - Assuming every square matrix is diagonal
A square matrix only requires equal number of rows and columns. In a diagonal matrix, all non-diagonal elements must be zero. - Confusing diagonal, scalar and identity matrices
Every identity matrix is a scalar matrix and every scalar matrix is a diagonal matrix, but the reverse is not always true. - Forgetting the properties of an identity matrix
In an identity matrix, all diagonal elements are 1 and all other elements are 0. - Mistakes in identifying elements
The notation a_{ij} represents the element in the i^{th} row and j^{th} column. - Ignoring order while checking equality of matrices
Two matrices can be equal only if they are of the same order. - Comparing matrices without checking corresponding elements
For equality of matrices, each corresponding element must be equal. - Confusing zero matrix with null value
A zero matrix is still a valid matrix in which all elements are zero.
Continue Learning
I hope these Matrices 3.1 NCERT Solutions helped you understand concepts like order of matrices and equality of matrices clearly. Do practice these questions regularly and try solving similar questions from other reference books and previous year questions (PYQs) as well.
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