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NCERT Solutions for Class 12 Maths Chapter 3 Matrices are provided here. Matrices is an important and powerful tool in mathematics and it's basically introduced to solve simultaneous linear equations. maths chapter 3 class 12 includes the type of matrices, operations of matrices, and many more concepts. These concepts are used to solve challenging problems of 3d geometry, and determinants Therefore matrices class 12 solutions become important to get command of concepts. The NCERT Solutions are prepared by expert team according the latest syllabus of CBSE 2023. Also, Students can refer matrices class 12 NCERT solutions which covers all the questions of NCERT Books for Class 12 Maths.
The practice from NCERT solutions for class 12 maths Chapter 3 Matrices is very important to score well in academics as well as in competitive exams. Matrices class 12 also includes exercises you can also solve to build hold on the concepts. All the questions in NCERT maths class 12 chapter 3 are explained in a detailed manner. You can also refer to class 12 maths ncert solutions chapter 3 matrices by clicking on the link NCERT solutions for Class 12.
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Matrix Definition and Properties:
A matrix is an ordered rectangular array of numbers or functions.
A matrix of order m × n consists of m rows and n columns.
The order of a matrix is written as m × n, where m is the number of rows and n is the number of columns.
A matrix is called a square matrix when m = n.
A diagonal matrix A = [a_{ij}]_{m×m} has a_{ij} = 0 when i ≠ j.
A scalar matrix A = [a_{ij}]_{n×n} has a_{ij} = 0 when i ≠ j, aij = k (where k is a constant)
when i = j.
An identity matrix A = [a_{ij}]n×n has a_{ij} = 1 when i = j and a_{ij} = 0 when i ≠ j.
A zero matrix contains all its elements as zero.
A column matrix is of the form [A]_{n × 1}.
A row matrix is of the form [A]_{1 × n}.
Equality of Matrices:
Two matrices A and B are equal (A = B) if they have the same order and a_{ij} = b_{ij} for all the corresponding values of i and j.
Operations on Matrices:
Matrix Addition:
If A = [a_{ij}]_{m × n} and B = [b_{ij}]_{m × n}, then A + B = [a_{ij} + b_{ij}]m × n.
Matrix Subtraction:
If A = [a_{ij}]_{m × n} and B = [b_{ij}]_{m × n}, then A - B = [a_{ij} - b_{ij}]_{m × n}.
Multiplication of a Matrix by Scalar:
Let A = [a_{ij}]_{m × n} be a matrix and k is a scalar, then kA is obtained by multiplying each element of A by the scalar k, i.e., kA = [ka_{ij}]_{m × n}.
Multiplication of Matrices:
Let A be an m × p matrix, and B be a p × n matrix. Their product AB is defined if the number of columns in A is equal to the number of rows in B. The resulting matrix is an m × n matrix, and the elements are calculated as follows: (AB)_{ij} = Σ(a_{i} * b_{j}), where the sum is taken over all values of p.
Transpose of a Matrix:
The transpose of a matrix A, denoted as AT, is obtained by interchanging its rows and columns.
Symmetric and Skew-Symmetric Matrices:
A matrix A is symmetric if A = AT (i.e., it is equal to its transpose).
A matrix A is skew-symmetric if AT = -A (i.e., the transpose of A is equal to the negative of A).
Elementary Operation or Transformation of a Matrix:
Elementary row operations include:
Interchanging any two rows.
Multiplying a row by a non-zero scalar.
Adding or subtracting a multiple of one row from another row.
Inverse of a Matrix by Elementary Operations:
You can find the inverse of a matrix using elementary row operations. If the matrix A is invertible, you can transform it into the identity matrix I through row operations on an augmented matrix [A | I], where I is the identity matrix of the same order as A. If this process is successful, the resulting matrix on the left will be I, and the matrix on the right will be the inverse of A.
Free download Matrices Class 12 Questions And Answers for CBSE Exam.
Matrices Class 12 Questions And Answers: Exercise 3.1
Question:1(i). In the matrix , write:
Answer:
(i) The order of the matrix = number of row number of columns .
Question 1(iii). In the matrix , write:
Write the elements a _{ 13 } , a _{ 21 } , a _{ 33 } , a _{ 24 } , a _{ 23 }
Answer:
(iii) An element implies the element in raw number i and column number j.
_{ }
Question 2. If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?
Answer:
A matrix has 24 elements.
The possible orders are :
.
If it has 13 elements, then possible orders are :
.
Question 3. If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
Answer:
A matrix has 18 elements.
The possible orders are as given below
If it has 5 elements, then possible orders are :
.
Question 4(i). Construct a 2 × 2 matrix, whose elements are given by:
Answer:
(i)
Each element of this matrix is calculated as follows
Matrix A is given by
Question 4(ii). Construct a 2 × 2 matrix, , whose elements are given by:
Answer:
A 2 × 2 matrix,
(ii)
Hence, the matrix is
Question 4(iii). Construct a 2 × 2 matrix, , whose elements are given by:
Answer:
(iii)
Hence, the matrix is given by
Question 5(i). Construct a 3 × 4 matrix, whose elements are given by:
Answer:
(i)
Hence, the required matrix of the given order is
Question 5(ii) Construct a 3 × 4 matrix, whose elements are given by:
Answer:
A 3 × 4 matrix,
(ii)
Hence, the matrix is
Question 6(i). Find the values of x, y and z from the following equations:
Answer:
(i)
If two matrices are equal, then their corresponding elements are also equal.
Question 6(ii) Find the values of x, y and z from the following equations:
Answer:
(ii)
If two matrices are equal, then their corresponding elements are also equal.
Solving equation (i) and (ii) ,
solving this equation we get,
Putting the values of y, we get
And also equating the first element of the second raw
,
Hence,
Question 6(iii) Find the values of x, y and z from the following equations
Answer:
(iii)
If two matrices are equal, then their corresponding elements are also equal
subtracting (2) from (1) we will get y=4
substituting the value of y in equation (3) we will get z=3
now substituting the value of z in equation (2) we will get x=2
therefore,
, and
Question 7. Find the value of a, b, c and d from the equation:
Answer:
If two matrices are equal, then their corresponding elements are also equal
Solving equation 1 and 3 , we get
Putting the value of a in equation 2, we get
Putting the value of c in equation 4 , we get
Question 8. is a square matrix, if
Answer:
A square matrix has the number of rows and columns equal.
Thus, for to be a square matrix m and n should be equal.
Option (c) is .
Question 9. Which of the given values of x and y make the following pair of matrices equal
Answer:
Given,
If two matrices are equal, then their corresponding elements are also equal
Here, the value of x is not unique, so option B is correct.
Question 10. The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:
Answer:
Total number of elements in a 3 × 3 matrix
If each entry is 0 or 1 then for every entry there are 2 permutations.
The total permutations for 9 elements
Thus, option (D) is correct.
Class 12 Maths Chapter 3 Question Answer: Exercise 3.2
Question 1(i) Let , ,
Answer:
(i) A + B
The addition of matrix can be done as follows
Question 1(iii) Let , ,
Answer:
(iii) 3A - C
First multiply each element of A with 3 and then subtract C
Question 2(ii). Compute the following:
Answer:
(ii) The addition operation can be performed as follows
Question 2(iii). Compute the following:
Answer:
(iii) The addition of given three by three matrix is performed as follows
Question 3(i). Compute the indicated products.
Answer:
(i) The multiplication is performed as follows
Question 3(ii). Compute the indicated products.
Answer:
(ii) the multiplication can be performed as follows
Question 3(iii). Compute the indicated products.
Answer:
(iii) The multiplication can be performed as follows
Question 3(iv). Compute the indicated products.
Answer:
(iv) The multiplication is performed as follows
Question 3(vi). Compute the indicated products.
Answer:
(vi) The given product can be computed as follows
Question 4. If , and , then compute (A+B) and (B-C). Also verify that A + (B - C) = (A + B) - C
Answer:
, and
Now, to prove A + (B - C) = (A + B) - C
(Puting value of from above)
Hence, we can see L.H.S = R.H.S =
Question 6. Simplify .
Answer:
The simplification is explained in the following step
the final answer is an identity matrix of order 2
Question 7(i). Find X and Y, if
Answer:
(i) The given matrices are
and
Adding equation 1 and 2, we get
Putting the value of X in equation 1, we get
Question 7(ii). Find X and Y, if
Answer:
(ii) and
Multiply equation 1 by 3 and equation 2 by 2 and subtract them,
Putting value of Y in equation 1 , we get
Question 10. Solve the equation for x, y, z and t, if
Answer:
Multiplying with constant terms and rearranging we can rewrite the matrix as
Dividing by 2 on both sides
Question 11. If , find the values of x and y.
Answer:
Adding both the matrix in LHS and rewriting
Adding equation 1 and 2, we get
Put the value of x in equation 2, we have
Question 12. Given , find the values of x, y, z and w.
Answer:
If two matrices are equal than corresponding elements are also equal.
Thus, we have
Put the value of x
Hence, we have
Question 14(i). Show that
Answer:
To prove:
Hence, the right-hand side not equal to the left-hand side, that is
Question 14(ii). Show that
Answer:
To prove the following multiplication of three by three matrices are not equal
Hence, i.e. .
Question 16. If prove that .
Answer:
First, find the square of matrix A and then multiply it with A to get the cube of matrix A
L.H.S :
Hence, L.H.S = R.H.S
i.e. .
Question 18. If and I is the identity matrix of order 2, show that
Answer:
To prove :
L.H.S :
R.H.S :
Hence, we can see L.H.S = R.H.S
i.e. .
Answer:
Let Rs. x be invested in the first bond.
Money invested in second bond = Rs (3000-x)
The first bond pays 5% interest per year and the second bond pays 7% interest per year.
To obtain an annual total interest of Rs. 1800, we have
(simple interest for 1 year )
Thus, to obtain an annual total interest of Rs. 1800, the trust fund should invest Rs 15000 in the first bond and Rs 15000 in the second bond.
Answer:
Let Rs. x be invested in the first bond.
Money invested in second bond = Rs (3000-x)
The first bond pays 5% interest per year and the second bond pays 7% interest per year.
To obtain an annual total interest of Rs. 1800, we have
(simple interest for 1 year )
Thus, to obtain an annual total interest of Rs. 2000, the trust fund should invest Rs 5000 in the first bond and Rs 25000 in the second bond.
Answer:
The bookshop has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books.
Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively.
The total amount the bookshop will receive from selling all the books:
The total amount the bookshop will receive from selling all the books is 20160.
Q21. The restriction on n, k and p so that PY + WY will be defined are:
(A)
Answer:
P and Y are of order and respectivly.
PY will be defined only if k=3, i.e. order of PY is .
W and Y are of order and respectivly.
WY is defined because the number of columns of W is equal to the number of rows of Y which is 3, i.e. the order of WY is
Matrices PY and WY can only be added if they both have same order i.e = implies p=n.
Thus, are restrictions on n, k and p so that PY + WY will be defined.
Option (A) is correct.
Question 22 Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k ,
respectively. Choose the correct answer in Exercises 21 and 22. If n = p , then the order of the matrix is:
(A) p × 2
(B) 2 × n
(C) n × 3
(D) p × n
Answer:
X has of order .
7X also has of order .
Z has of order .
5Z also has of order .
Mtarices 7X and 5Z can only be subtracted if they both have same order i.e = and it is given that p=n.
We can say that both matrices have order of .
Thus, order of is .
Option (B) is correct.
Class 12 Maths Chapter 3 Question Answer: Exercise 3.3
Question 1(i). Find the transpose of each of the following matrices:
Answer:
The transpose of the given matrix is
Question 1(ii). Find the transpose of each of the following matrices:
Answer:
interchanging the rows and columns of the matrix A we get
Question 1(iii) Find the transpose of each of the following matrices:
Answer:
Transpose is obtained by interchanging the rows and columns of matrix
Question 2(i). If and , then verify
Answer:
and
L.H.S :
R.H.S :
Thus we find that the LHS is equal to RHS and hence verified.
Question 2(ii). If and , then verify
Answer:
and
L.H.S :
R.H.S :
Hence, L.H.S = R.H.S. so verified that
.
Question 4. If and , then find
Answer:
:
Transpose is obtained by interchanging rows and columns and the transpose of A+2B is
Question 5(i) For the matrices A and B, verify that , where
Answer:
,
To prove :
Hence, L.H.S =R.H.S
so it is verified that .
Question 5(ii) For the matrices A and B, verify that , where
Answer:
,
To prove :
Heence, L.H.S =R.H.S i.e. .
Question 6(i). If , then verify that
Answer:
By interchanging rows and columns we get transpose of A
To prove:
L.H.S :
Question 6(ii). If , then verify that
Answer:
By interchanging columns and rows of the matrix A we get the transpose of A
To prove:
L.H.S :
Question 7(i). Show that the matrix is a symmetric matrix.
Answer:
the transpose of A is
Since, so given matrix is a symmetric matrix.
Question 7(ii) Show that the matrix is a skew-symmetric matrix.
Answer:
The transpose of A is
Since, so given matrix is a skew-symmetric matrix.
Question 9. Find and , when
Answer:
the transpose of the matrix is obtained by interchanging rows and columns
Question 10(i). Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
Answer:
Let
Thus, is a symmetric matrix.
Let
Thus, is a skew symmetric matrix.
Represent A as sum of B and C.
Question:10(ii). Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
Answer:
Let
Thus, is a symmetric matrix.
Let
Thus, is a skew-symmetric matrix.
Represent A as the sum of B and C.
Question 10(iii). Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
Answer:
Let
Thus, is a symmetric matrix.
Let
Thus, is a skew-symmetric matrix.
Represent A as the sum of B and C.
Question 10(iv). Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
Answer:
Let
Thus, is a symmetric matrix.
Let
Thus, is a skew-symmetric matrix.
Represent A as the sum of B and C.
Question 11 Choose the correct answer in the Exercises 11 and 12.
If A, B are symmetric matrices of same order, then AB – BA is a
(A) Skew symmetric matrix
(B) Symmetric matrix
(C) Zero matrix
(D) Identity matrix
Answer:
If A, B are symmetric matrices then
and
we have,
Hence, we have
Thus,( AB-BA)' is skew symmetric.
Option A is correct.
NCERT solutions for class 12 maths chapter 3 Matrices: Exercise 3.4
Answer:
Use the elementary transformation we can find the inverse as follows
Answer:
Thus we have obtained the inverse of the given matrix through elementary transformation
Answer:
Using elementary transformations
.
Answer:
Answer:
.
Thus the inverse of matrix A is obtained.
Answer:
Use the elementary transformation
.
Answer:
.
Thus the inverse of matrix A is obtained using elementary transformation.
Answer:
Thus using elementary transformation inverse of A is obtained as
.
Answer:
Thus using elementary transformation the inverse of A is obtained as
.
Answer:
.
Thus the inverse of A is obtained using elementary transformation.
Answer:
thus the inverse of matrix A is
.
Answer:
Hence, we can see all the zeros in the second row of the matrix in L.H.S so does not exist.
Answer:
so the inverse of matrix A is
.
Answer:
Hence, we can see all upper values of matirix are zeros in L.H.S so does not exists.
Answer:
Thos the Inverse of A is
. .
Answer:
and
and
and
Thus the inverse of three by three matrix A is
. .
Answer:
and
Thus the inverse of A is obtained as
. .
Question:18 Matrices A and B will be inverse of each other only if
Answer:
We know that if A is a square matrix of order n and there is another matrix B of same order n, such that , then B is inverse of matrix A.
In this case, it is clear that A is inverse of B.
Hence, m atrices A and B will be inverse of each other only if .
Option D is correct .
NCERT solutions for class 12 maths chapter - 3 Matrices: Miscellaneous exercise
Question:1 Let , show that , where I is the identity matrix of order 2 and .
Answer:
Given :
To prove :
For n=1,
The result is true for n=1.
Let result be true for n=k,
Now, we prove that the result is true for n=k+1,
Put the value of in above equation,
Hence, the result is true for n=k+1.
Thus, we have where , .
Question 2. If then show that , .
Answer:
Given :
To prove:
For n=1, we have
Thus, the result is true for n=1.
Now, take n=k,
For, n=k+1,
Thus, the result is true for n=k+1.
Hence, we have , where .
Question 3. If , then prove that , where n is any positive integer.
Answer:
Given :
To prove:
For n=1, we have
Thus, result is true for n=1.
Now, take result is true for n=k,
For, n=k+1,
Thus, the result is true for n=k+1.
Hence, we have , where .
Question 4. If A and B are symmetric matrices, prove that is a skew symmetric matrix.
Answer:
If A, B are symmetric matrices then
and
we have,
Hence, we have
Thus,( AB-BA)' is skew symmetric.
Question 5. Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Answer:
Let be a A is symmetric matrix , then
Consider,
Replace by
i.e.
Thus, if A is a symmetric matrix than is a symmetric matrix.
Now, let A be a skew-symmetric matrix, then .
Replace by - ,
i.e. .
Thus, if A is a skew-symmetric matrix then is a skew-symmetric matrix.
Hence, the matrix B′AB is symmetric or skew-symmetric according to as A is symmetric or skew-symmetric.
Question 6. Find the values of x , y , z if the matrix satisfy the equation
Answer:
Thus equating the terms elementwise
Question 10(a) A manufacturer produces three products x, y, z which he sells in two markets.
Annual sales are indicated below:
Market Products
I 10,000 2,000 18,000
II 6,000 20,000 8,000
Answer:
The unit sale prices of x, y and z are ` 2.50, ` 1.50 and ` 1.00, respectively.
The total revenue in the market I with the help of matrix algebra can be represented as :
The total revenue in market II with the help of matrix algebra can be represented as :
Hence, total revenue in the market I is 46000 and total revenue in market II is 53000.
Question 10(b). A manufacturer produces three products x, y, z which he sells in two markets.
Annual sales are indicated below:
Market Products
I 10,000 2,000 18,000
II 6,000 20,000 8,000
Answer:
The unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively.
The total cost price in market I with the help of matrix algebra can be represented as :
Total revenue in the market I is 46000 , gross profit in the market is
The total cost price in market II with the help of matrix algebra can be represented as :
Total revenue in market II is 53000, gross profit in the market is
Question 11. Find the matrix X so that
Answer:
The matrix given on R.H.S is matrix and on LH.S is matrix.Therefore, X has to be matrix.
Let X be
Taking,
Hence, we have
Matrix X is .
Answer:
A and B are square matrices of the same order such that ,
To prove : ,
For n=1, we have
Thus, the result is true for n=1.
Let the result be true for n=k,then we have
Now, taking n=k+1 , we have
Thus, the result is true for n=k+1.
Hence, we have , .
To prove:
For n=1, we have
Thus, the result is true for n=1.
Let the result be true for n=k,then we have
Now, taking n=k+1 , we have
Thus, the result is true for n=k+1.
Hence, we have and for all .
Question 13 Choose the correct answer in the following questions:
Answer:
Thus we obtained that
Option C is correct.
Question 14. If the matrix A is both symmetric and skew-symmetric, then
(A) A is a diagonal matrix
(B) A is a zero matrix
(C) A is a square matrix
(D) None of these
Answer:
If the matrix A is both symmetric and skew-symmetric, then
and
Hence, A is a zero matrix.
Option B is correct.
Question 15. If A is square matrix such that , then is equal to
Answer:
A is a square matrix such that
(Replace by )
Hence, we have
Option C is correct.
If you are interested in Matrices Class 12 NCERT Solutions exercises then these are listed below.
Matrices Class 12 NCERT Solutions Exercise 3.1
Matrices Class 12 NCERT Solutions Exercise 3.2
Matrices Class 12 NCERT Solutions Exercise 3.3
Matrices Class 12 NCERT Solutions Exercise 3.4
Matrices Class 12 NCERT Solutions Miscellaneous Exercise
Matrix is an array of numbers. Matrix is a mode of representing data to ease calculation and it is one of the most important tools of mathematics because matrices simplify our work to a great extent when compared with other straight forward methods. Exercises are given in chapter 3 maths class 12 students should refer for practice. Matrices class 12 solutions also provided by careers360’s expert team if you are facing problems then you can refer to it as well.
Matrices are used as a representation of the coefficients in the system of linear equations, electronic spreadsheet programs, also used in business and science. For the students to understand NCERT class 12 maths solutions chapter 3 in a better way, a total of 28 solved examples are given to practice more, at the end of the chapter, 15 questions are given in the miscellaneous exercise. For command on concepts you can uses these practice problems of matrices class 12.
3.1 Introduction
3.2 Matrix
3.2.1 Order of a matrix
3.3 Types of Matrices
3.3.1 Equality of matrices
3.4 Operations on Matrices
3.4.1 Addition of matrices
3.4.2 Multiplication of a matrix by a scalar
3.4.3 Properties of matrix addition
3.4.4 Properties of scalar multiplication of a matrix
3.4.5 Multiplication of matrices
3.4.6 Properties of multiplication of matrices
3.5. Transpose of a Matrix
3.5.1 Properties of the transpose of the matrices
3.6 Symmetric and Skew-Symmetric Matrices
3.7 Elementary Operation (Transformation) of a Matrix
3.8 Invertible Matrices
3.8.1 Inverse of a matrix by elementary operations
Topics of NCERT Class 12 Maths Chapter Matrices
The main topics covered in maths chapter 3 class 12 are:
Matrix
It is an ordered representation of numbers and functions, known as elements in a rectangular array. In this class 12 NCERT topics discuss concepts related to the order of a matrix, elements in raw, and columns of a matrix. there are good quality questions in matrix class 12 solutions.
Type of matrices
This ch 1 maths class 12 concerns different types of matrices like column matrix, raw matrix, square matrix, diagonal matrix, scalar matrix, identity matrix, zero matrix, etc. In this ch 3 maths class 12 also discuss comprehensively conditions of equality of matrix. To get command on these concepts you can refer to NCERT solutions for class 12 maths chapter 3.
Operations on matrices
This ch 3 maths class 12 also includes concepts of addition of matrices, multiplication of a matrix by a scalar, negative matrix, difference of matrices. maths class 12 chapter 3 also contains properties of matrix addition that include commutative leas, associative laws, the existence of an additive identity, the existence of additive inverse. Properties of scalar multiplication of matrix elaborated in this chapter. You can refer to class 12 NCERT solutions for questions about these concepts.
multiplication of matrix
Properties of multiplication of matrix that includes associative law, distributive laws, the existence of multiplicative identity. to get command of these concepts you can go through the NCERT solution for class 12 maths chapter 3.
transpose of a matrix
properties of transpose of the matrix are discussed in maths class 12 chapter 3. matrix class 12 solutions include quality questions to understand the concepts.
ch 3 maths class 12 also discuss in detail the concepts of symmetric and skew symmetric matrix, elementary operations (transformation) of a matrix. for questions on these concepts, you can browse NCERT solutions for class 12 chapter 3.
In class 12 NCERT chapter you get an idea of the inverse of a matrix by elementary operation. class 12 NCERT solutions include problems related to these concepts
Topics enumerated in class 12 NCERT are very important and students are suggested to go through all the concepts discussed in the topics. Questions related to all the above topics are covered in the NCERT solutions for class 12 maths chapter 3
Also read,
Chapter 1 | Relations and Functions |
Chapter 2 | Inverse Trigonometric Functions |
Chapter 3 | Matrices |
Chapter 4 | Determinants |
Chapter 5 | Continuity and Differentiability |
Chapter 6 | Application of Derivatives |
Chapter 7 | Integrals |
Chapter 8 | Application of Integrals |
Chapter 9 | Differential Equations |
Chapter 10 | Vector Algebra |
Chapter 11 | Three Dimensional Geometry |
Chapter 12 | Linear Programming |
Chapter 13 | Probability |
NCERT Class 12 Maths solutions chapter 3 covers all the questions given in the textbook. Make sure you have gone through all the important concepts of NCERT class 12th maths Matricesbefore solving the questions.
Firstly try to solve the questions on your own and then check the NCERT class 12 maths chapter 3 solutions.
Along with the NCERT Solutions for Class 12 Maths Chapter 3 Matrices, solve the previous year questions as well.
NCERT Class 12 maths chapter 3 solutions pdf download will be made available soon. Till then you can save this webpage instead of NCERT class 12 maths chapter 3 pdf to practice questions offline.
NCERT solutions are not only important when you are stuck while solving the problems but you will get to know how to answer in the board exam in order to get good marks in the board exam. these class 12 maths ch 3 question answer also make you understand new concepts or in depth understanding of topics and can solve you doubts. in this way ncert solutions becomes important so you can refer ncert textbooks, ncert exercises and ncert solutions.
Matrices, order of a matrix, types of matrices, equality of matrices, operations like addition multiplication on matrices, symmetric and skew-symmetric matrices are the important topics in this chapter. these topics are very important because of use in other supplementary topics like 3d geometry.
Yes, Class 12 Matrices solutions definitely improve the performance on the board exam. It's recommended that students begin by tackling simpler problems before moving on to more challenging ones. Class 12 matrices NCERT solutions will enable them to identify areas where they need improvement. Through repeated practice on their weaker concepts, students can strengthen their skills and perform well on the board exams. Additionally, short-cut tips are provided to assist students in finding easier ways to solve complex problems.
The topic algebra which contains two topics matrices and determinants which has 13 % weightage in the maths CBSE 12th board final examination. students can list out topics according to respective weightage and channelise their energy according to priority of the topics.
Mathematics often has simple chapters that can be enjoyable to study once understood, and matrices are one example. NCERT Class 12 maths matrices focus on the following main topics: matrices, types of matrices, operations on matrices, transpose of a matrix, symmetric and skew symmetric matrices, elementary operations on matrices, and invertible matrices. The material is presented in a straightforward manner to help students achieve good grades in the board exams, regardless of their intelligence. For ease, students can study matrices class 12 ncert pdf both online and offline.
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Waitlist: Many schools maintain waitlists after their initial application rounds close. If a student who secured a seat decides not to join, the school might reach out to students on the waitlist. So, even if the application deadline has passed, it might be worth inquiring with schools you're interested in if they have a waitlist and if they would consider adding you to it.
Schools with ongoing admissions: Some schools, due to transfers or other reasons, might still have seats available even after the main admission rush. Reach out to the schools directly to see if they have any open seats in 10th grade.
Consider other good schools: There might be other schools in your area that have a good reputation but weren't on your initial list. Research these schools and see if they have any seats available.
Focus on excelling in your current school: If you can't find a new school this year, focus on doing well in your current school. Maintain good grades and get involved in extracurricular activities. This will strengthen your application for next year if you decide to try transferring again.
Best CBSE schools in Delhi: Click Here
In India, the design and coding fields offer exciting career options that can leverage your interest in both. Here's how you can navigate this path:
Choosing Your Stream:
Graphic Design Focus: Consider a Bachelor's degree in Graphic Design or a design diploma. Build a strong portfolio showcasing your creative skills. Learn the basics of HTML, CSS, and JavaScript to understand web development better. Many online resources and bootcamps offer these introductory courses.
Coding Focus: Pursue a Computer Science degree or a coding bootcamp in India. These programs are intensive but can equip you with strong coding skills quickly. While building your coding prowess, take online courses in graphic design principles and UI/UX design.
Engineering Subjects (for a Degree):
Information Technology (IT): This offers a good mix of web development, networking, and database management, all valuable for web design/development roles.
Human-Computer Interaction (HCI): This is a specialized field that bridges the gap between design and computer science, focusing on how users interact with technology. It's a perfect choice if you're interested in both aspects.
Here's why 2025 is more likely:
Looking Ahead (2025 Admissions):
Yes, scoring above 99.9 percentile in CAT significantly increases your chances of getting a call from IIM Bangalore, with your academic background. Here's why:
High CAT Score: A score exceeding 99.9 percentile is exceptional and puts you amongst the top candidates vying for admission. IIM Bangalore prioritizes CAT scores heavily in the shortlisting process.
Strong Academics: Your 96% in CBSE 12th and a B.Tech degree demonstrate a solid academic foundation, which IIM Bangalore also considers during shortlisting.
However, the shortlisting process is multifaceted:
Here's what you can do to strengthen your application:
Focus on WAT and PI: If you receive a shortlist, prepare extensively for the Written Ability Test (WAT) and Personal Interview (PI). These stages assess your communication, soft skills, leadership potential, and suitability for the program.
Work Experience (if applicable): If you have work experience, highlight your achievements and how they align with your chosen IIM Bangalore program.
Overall, with a stellar CAT score and a strong academic background, you have a very good chance of getting a call from IIM Bangalore. But remember to prepare comprehensively for the other stages of the selection process.
hello,
Yes you can appear for the compartment paper again since CBSE gives three chances to a candidate to clear his/her exams so you still have two more attempts. However, you can appear for your improvement paper for all subjects but you cannot appear for the ones in which you have failed.
I hope this was helpful!
Good Luck
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