Matrices Class 12th Notes - Free NCERT Class 12 Maths Chapter 3 Notes

Q: What is the Transpose of a Matrix?

Let be any matrix of order m x n then we can get hen transpose of matrix A by interchanging its rows with columns. i.e. A T or .

Matrices Class 12th Notes - Free NCERT Class 12 Maths Chapter 3 Notes - Download PDF

Edited By Ravindra Pindel | Updated on Apr 23, 2022 03:07 PM IST

NCERT Class 12 Maths Chapter 3 Matrices Notes- Matrices was introduced to solve simultaneous linear equations. It has an important application in most scientific fields like classical mechanics, optics, electromagnetism, quantum mechanics, quantum electrodynamics and, many more concepts of physics. In this article, you will get NCERT Class 12 Maths Chapter 3 Notes which are very important for the CBSE board exam as well as competitive exams. These notes are prepared by experts who know how best to prepare for board exams. Students are advised to solve all the NCERT problems on their own. You can take help from Matrices Class 12 Notes, which will give a short description of the topics covered in this chapter. Here, you can also check NCERT Book

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Matrix:-

A rectangular array of numbers horizontal lines and vertical lines is called a matrix. Let's there are 'm' horizontal lines and 'n' vertical lines then the order of the matrix is m by n which is written as m x n matrix.

$A=\left[\begin{array}{ccccc} a_{11} & a_{12} & a_{13} & \ldots & a_{\ln } \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2 n} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{m 1} & a_{m 2} & a_{m 3} & \ldots & a_{m n} \end{array}\right]$ aij belongs to the ith row and jth column and is called the (i, j) th element of the matrix

$A=\left[\begin{array}{ccc} 1 & 2 & 3 \\ 0 & -1 & 9 \end{array}\right]$

$a_{11}=1, a_{12}=2, a_{13}=3, a_{21}=0, a_{22}=-1, a_{23}=9$

SPECIAL TYPE OF MATRICES :

Row Matrix (Row vector) :

Matrix that has only one row.

$\mathrm{A}=\left[\mathrm{a}_{11}, \mathrm{a}_{12}, \ldots \ldots \ldots \ldots \mathrm{a}_{1 \mathrm{n}}\right]$

Column Matrix (Column vector) :

Matrix that has only one column.

$A=\left[\begin{array}{c} a_{11} \\ a_{21} \\ . \\ .\\ .\\ a_{m 1} \end{array}\right]$

Zero or Null Matrix :

Matrix whose all entries are zero.

$\mathrm{A}=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right]$

Horizontal Matrix:

A matrix that has more columns than the number of rows.

$\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 5 & 1 & 1 \end{array}\right]$

Vertical Matrix :

A matrix that has more rows than the number of columns.

$A= \left[\begin{array}{ll} 2 & 5 \\ 1 & 1 \\ 3 & 6 \\ 2 & 4 \end{array}\right]$

Square Matrix :

In a square matrix number of rows are equal to the number of columns.

$A=\left[\begin{array}{rrr} -5 & -8 & 0 \\ 3 & 5 & 0 \\ 1 & 2 & -1 \end{array}\right]$

Trace of Matrix :

The sum of the elements of a square matrix A lying along the principal diagonal is called the trace of A.

$A=\left[\begin{array}{ccccc} a_{11} & a_{12} & a_{13} & \ldots & a_{\ln } \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2 n} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{n 1} & a_{n 2} & a_{n 3} & \ldots & a_{n n} \end{array}\right]$

$\operatorname{tr}(\mathrm{A})=\sum_{i=1}^{\mathrm{a}} \mathrm{a}_{\mathrm{ii}}=\mathrm{a}_{11}+\mathrm{a}_{22}+\ldots \ldots \mathrm{a}_{\mathrm{nn}}$

Properties of trace of a matrix :

Let A and B are two matrices.

$\\ (i) \operatorname{tr}(\lambda, A)=\lambda \operatorname{tr}(A) \\ (ii) \operatorname{tr}(\mathrm{A}+\mathrm{B})=\operatorname{tr}(A) +tr(B) \\ (iii) \operatorname{tr}(\mathrm{AB})=\operatorname{tr}(\mathrm{BA})$

Equality of Matrices:

Matrices A and B are equal if '

(i) Order of matrix A are equal to order of matrix B.

(ii) $a_{i j}=b_{i j}$ for each pair

Addition of Matrices:

If matrix A and matrix are of same order then addition of matix A and B given by

A + B = [aij + bij]

Properties-

( a ) Addition of matrices is commutative : i.e. A + B = B + A

( b ) Matrix addition is associative : (A + B) + C = A + (B + C)

Additive inverse : Let two matrices A and B are such that A + B = O = B + A, then B is called additive inverse of A.

Multiplication of Matrix by a Scalar:

- If A and B are two matrices of the same order and 'k' be a scalar then

k(A + B) = kA + kB.

- If p and q are two scalars and 'A' is a matrix, then

(p+q )A = pA + qA

- If p and q are two scalars and 'A' is a matrix, then

(pq )A =p (qA) =q (pA)

Multiplication of Matrices:

Let A be a matrix of order m x n and B be a matrix of order p x q, then the matrix multiplication AB is possible if and only if n = p. The multiplication of AB is given by

$(\mathrm{AB})_{ij}=\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{a}_{\mathrm{ir}} \mathrm{b}_{\mathrm{rj}}$

Properties of Matrix Multiplication:

(i) Matrix multiplication is not commutative : i.e. $AB \neq BA$

( ii ) Matrix Multiplication Is Associative : If A, B & C are conformable for the product AB & BC, then (AB) C = A(BC)

( ii ) Matrix Multiplication Is Distributivity : A(B+C)=A B+A C and (A+B) C=A C+B C

Transpose of Matrix:

Let $A = [a_{ij}]$ be any matrix of order m x n. Then transpose of matrix A i.e. A^T or $A' = [a_{ji}]$

Properties of transpose :

$(\mathrm{A}+\mathrm{B})^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}+\mathrm{B}^{\mathrm{T}}$
$(\mathrm{A} \mathrm{B})^{\mathrm{T}}=\mathrm{B}^{\mathrm{T}} \mathrm{A}^{\mathrm{T}}$
$\left(\mathrm{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\mathrm{A}$
$(\mathrm{kA})^{\mathrm{T}}=\mathrm{kA}^{\mathrm{T}}$ where 'k' is a scaler

Orthogonal Matrix:

A square matrix is said to be an orthogonal matrix if

$\mathrm{A} \mathrm{A}^{\mathrm{T}}=\mathrm{I}$

Symmetric matrix :

A square matrix $A = [ aij ]$ is said to be a symmetric matrix if

$a_{ij} = a_{ji}$ for each element in the matrix

Skew symmetric matrix :

A square matrix $A = [ aij ]$ is said to be a skew-symmetric matrix if

$a_{ij} = -a_{ji}$ for each element in the matrix

Properties of symmetric & skew-symmetric matrix :

Matrix A is symmetric if $A^T = A$
Matrix A is skew-symmetric if $A^T = -A$
The sum of two symmetric matrices is a symmetric matrix and the sum of two skew-symmetric matrices is a skew-symmetric matrix.
If A & B are symmetric matrices then AB + BA is a symmetric matrix and AB – BA is a skew-symmetric matrix.

Elementary operations of a matrix:

Following elementary operations are allowed on matrix.

$\\ \mathrm{R_i} \leftrightarrow \mathrm{R_j} \ or \ \mathrm{C_i}\leftrightarrow \mathrm{C_j}$
$\mathrm{R}_{i} \rightarrow k \mathrm{R}_{i} \ or \ \mathrm{C}_{i} \rightarrow k \mathrm{C}_{i}$
$\mathrm{R}_{i} \rightarrow \mathrm{R}_{\mathrm{j}}+k \mathrm{R}_{i} \ or \ \mathrm{C}_{i} \rightarrow \mathrm{C}_{i}+k \mathrm{C}_{j}$

Inverse of a Matrix:

If A and B are two square matrices such that $AB = BA = I$ then B is the inverse matrix of A and is denoted by A^–1 and A is the inverse of B.

Note- It is necessary that matrices A and B should be square matrices of the same order to be invertible matrices.

- If B is the inverse of A, then A is also the inverse of B

NCERT Class 12 Notes Chapter Wise.

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NCERT Books and Syllabus

NCERT Book for Class 12

NCERT Syllabus for Class 12

Important points to Note:-

NCERT problems are very important in order to perform well in the exam. Students must solve all the NCERT problems including miscellaneous exercises on their own.
Students are advised to go through NCERT Class 12 Maths Chapter 3 Notes before solving the questions.
You can take help from these Matrices Class 12 Notes for quick revision before the exam.

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Frequently Asked Questions (FAQs)

1. What is the Transpose of a Matrix?

Let $A = [a_{ij}]$ be any matrix of order m x n then we can get hen transpose of matrix A by interchanging its rows with columns. i.e. A^T or $A' = [a_{ji}]$ .

2. Does CBSE provides the revision notes for NCERT Class 12?

No, CBSE doesn't provide any short notes or revision notes for any class.

3. Which is the best book for CBSE Class 12 Maths ?

NCERT textbook is the most important book for CBSE Class 12 Maths. You can also refer Maths books by RS Aggarwal and RD Sharma.

4. What is the weightage of the chapter Matrices for CBSE board exam ?

The total weightage of Matrices is 6 marks in the final board exam.

5. Which are the most difficult chapters of NCERT Class 12 Maths syllabus?

Some students consider Probability and Integration are the most difficult chapter in the CBSE Class 12 Maths but with rigorous practice, students can get command on these chapters very easily.

6. How does the NCERT Notes are helpful in the board exam ?

NCERT notes are helpful in understanding some important concepts and can be used to revise important concepts.

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Popular Questions

A block of mass 0.50 kg is moving with a speed of 2.00 ms^-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1) $0.34\; J$	Option 2) $0.16\; J$
Option 3) $1.00\; J$	Option 4) $0.67\; J$

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up ? Fat supplies 3.8×10⁷J of energy per kg which is converted to mechanical energy with a 20% efficiency rate. Take g = 9.8 ms⁻² :

Option 1)

2.45×10⁻³ kg

Option 2)

6.45×10⁻³ kg

Option 3)

9.89×10⁻³ kg

Option 4)

12.89×10⁻³ kg

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Option 1) $2,000 \; J - 5,000\; J$	Option 2) $200 \, \, J - 500 \, \, J$
Option 3) $2\times 10^{5}J-3\times 10^{5}J$	Option 4) $20,000 \, \, J - 50,000 \, \, J$

A particle is projected at 60⁰ to the horizontal with a kinetic energy $K$ . The kinetic energy at the highest point

Option 1) $K/2\,$	Option 2) $\; K\;$
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Option 1) $11.2\, L\, H_{2(g)}$ at STP is produced for every mole $HCL_{(aq)}$ consumed	Option 2) $6L\, HCl_{(aq)}$ is consumed for ever $3L\, H_{2(g)}$ produced
Option 3) $33.6 L\, H_{2(g)}$ is produced regardless of temperature and pressure for every mole Al that reacts	Option 4) $67.2\, L\, H_{2(g)}$ at STP is produced for every mole Al that reacts .

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Option 1) 0.02	Option 2) 3.125 × 10^-2
Option 3) 1.25 × 10^-2	Option 4) 2.5 × 10^-2

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Option 1) decrease twice	Option 2) increase two fold
Option 3) remain unchanged	Option 4) be a function of the molecular mass of the substance.

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Option 1) Molality	Option 2) Weight fraction of solute
Option 3) Fraction of solute present in water	Option 4) Mole fraction.

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Option 1) twice that in 60 g carbon	Option 2) 6.023 × 10²²
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A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t²) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m² , the number of rotations made by the pulley before its direction of motion if reversed, is