### Featured Posts

### Chapter 3 Matrix Class XII : Basic Concepts

- Get link
- Other Apps

__Class 12 Chapter 3 Matrices__

*Complete explanation, basic concepts and formulas based on matrices. Matrix method of arranging the terms , addition, subtraction, multiplication , and transpose of the matrices.*

**Solution of Important questions of NCERT Chapter 3 Matrices**

__Basic Concepts Based on Chapter 3 (Matrix) Class 12__

**Definition :**

**A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.**

**Order of a Matrix:**

**A matrix having m rows and n columns is called a matrix of order m x n**

**In general a matrix of order m x n can be arranged in a rectangular array as**

**\[\begin{bmatrix} a_{11} & a_{12} & a_{13} &...... & a_{1n} \\ a_{21} & a_{22} & a_{23} &...... & a_{2n} \\ a_{31} &a_{32} & a_{33} &...... & a_{3n} \\ ....& .... & ... & ...... & .... \\ ....& .... & ... & ...... & .... \\ a_{m1}&a_{m2} &a_{m3} & ...... &a_{mn} \end{bmatrix}_{m\times n}\]**

**a**

_{11}means an element in the first row and first column, a_{23}means an element in the second row and in the third column and so on.

**In short a matrix can be written as A = [a**

_{ij}]_{m x n}, where a_{ij}represent the elements of the matrix, m is the number rows and n is the number of columns.

__Types of Matrices__

**Column Matrix :**

**If the matrix has only one column then it is called column matrix.**

**\[\begin{bmatrix} a\\b \\c \end{bmatrix}\]**

**Row Matrix : If a matrix has only one row then it is called a row matrix.**

**\[\begin{bmatrix} a & b &c \end{bmatrix}\]**

**Square Matrix :**

**A matrix in which number of rows and columns are equal is called a square matrix. For Example:**

**\[\begin{bmatrix} a &b \\c &d \end{bmatrix}_{2\times 2}\; \; \; or\; \; \; \begin{bmatrix} a &b &c \\d & e &f \\ 1& 2 &3 \end{bmatrix}_{3\times 3}\]**

**Diagonal Matrix :**

**A square matrix is said to be diagonal matrix if its all non diagonal elements are zero. For example :**

**\[\begin{bmatrix} a &0 &0 \\ 0 &b &0 \\ 0 & 0 & c \end{bmatrix}\]**

**Scalar Matrix :**

**A diagonal matrix is said to be scalar matrix if its all diagonal elements are equal. For Example :**

\[\begin{bmatrix} 5 &0 &0 \\ 0 &5 &0 \\ 0 & 0 & 5 \end{bmatrix}\]

**Identity Matrix :**

**A diagonal matrix is said to be identity matrix if its all diagonal elements are 1. Example :**

**\[\begin{bmatrix} 1 &0 &0 \\ 0 &1 &0 \\ 0 & 0 & 1 \end{bmatrix}\]**

**Zero Matrix :**

**A square matrix is said to be zero matrix if its all elements are zero.**

**\[O=\begin{bmatrix} 0 &0 &0 \\ 0 &0 &0 \\ 0 & 0 & 0 \end{bmatrix}\]**

**Equal Matrices :**

**Two matrices A and B are said to be equal if**

**a) Matrices A and B are of same order**

**b) Corresponding elements of matrix A and B are equal.**

__Addition of two matrices :__

**Addition of two matrices is possible if they are of same order**

**Addition of two matrices is done by adding their corresponding elements.**

**Example : \[\begin{bmatrix} a &b &c \\ x& y & z\\ p & q & r \end{bmatrix}+\begin{bmatrix} 1&2 &3 \\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}=\begin{bmatrix} a+1&b+2 &c+3 \\ x+4 & y+5 & z+6\\ p+7 & q+8 & r+9 \end{bmatrix}\]**

**Addition of two matrices is not possible if they are not of same order.**

__Multiplication of a matrix with a scalar__

**Let A is a matrix and k is a scalar, then multiplication of scalar k with matrix A is obtained by multiplying each element of matrix A by a scalar k.**

**\[If\; matrix\; A=\begin{bmatrix} x &y &z \\ p & q &r \\ a&b &c \end{bmatrix} \; and \; scalar\; k=2,\; then\] \[2A=2\begin{bmatrix} x &y &z \\ p & q &r \\ a&b &c \end{bmatrix}=\begin{bmatrix} 2x &2y &2z \\ 2p & 2q &2r \\ 2a & 2b &2c \end{bmatrix}\]**

__Properties of matrix addition__

**If A and B are two matrices then**

**Commutativity**

**Matrix addition holds the commutative law : A + B = B + A**

**Associative law**

**Matrix addition holds the associative law : (A + B) + C = A + (B + C)**

**Existence of Identity : If zero matrix O is added to any matrix A then matrix A is remain unchanged. A + O = O + A = A ⇒ O is the identity matrix in Matrix addition.**

**Existence of additive inverse:**

**If A is a matrix then A + (-A) = (-A) + A = O, then -A is said to be the Additive inverse of matrix A.**

__Multiplication of matrices__

**Product of two matrices A and B is defined if number of columns of first matrix A is equal to the number of rows of matrix B**

**Example :**

**If A and B are two matrices such that:**

\[A=\left [ a_{ij} \right ]_{m\times n}\; \; and\; \; B=\left [ b_{ij} \right ]_{p\times q}\]

**Then product AB exists if n = p and the order of the resulting matrix AB is m x q**

**Note :**

**If product AB is defined then product BA need not to be defined**

**Product AB and BA both are defined if A and B both are the square matrices of same order.**

**Example : If A and B are two matrices such that :**

**\[A = \begin{bmatrix} 2 &4 &-3 \\ 1 &4 & 3 \end{bmatrix}\: \: and\: \: B=\begin{bmatrix} 2 &-3 \\ 4 &5 \\ 1 & 2 \end{bmatrix}\]**

**Order of matrix A is 2 x 3, and the order of matrix B = 3 x 2.**

**Number of columns of first matrix is equal to the number of rows of second matrix.**

**Therefore product of A and B is defined. And the order of the resulting matrix become 2 x 2**

\[AB=\begin{bmatrix} 2\times 2+4\times4+(-3)\times1 &\; \; \; \; \; 2\times-3+4\times5+(-3)\times2 \\ 1\times2+4\times4+3\times1&1\times-3+4\times5+3\times2 \end{bmatrix}\]

\[AB=\begin{bmatrix} 4+16-3 &\; \; -6+20-6 \\ 2+16+3 &\; \; -3+20+6 \end{bmatrix}\]

**\[AB=\begin{bmatrix} 17 &\; \; 8 \\ 21 &\; \; 23 \end{bmatrix}\]**

__Properties of matrix multiplication.__

**i) Matrix multiplication is not commutative.**

**⇒ AB ≠ BA**

**ii) If A and B are the diagonal matrices of the same order then multiplication of the matrices is commutative. In this case we can write AB = BA**

**iii) Matrix multiplication is associative**

**⇒**

**(AB)C = A(BC)**

**iv) Matrix multiplication holds the Distributive property**

**A(B + C) = AB + AC or (A+B)C = AC + BC**

**Existence of multiplicative identity :**

**For every square matrix there exists an identity matrix I of the same order such that AI = IA = A,**

**⇒**

**I is the identity matrix for A.**

**Example**

**For a square matrix of 2, I is the identity matrix and is given by \[I=\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}=I_{3}\]**

**For a square matrix of 3, I is the identity matrix and is given by**

**\[I=\begin{bmatrix} 1 &0 &0 \\ 0&1 &0 \\ 0& 0 &1 \end{bmatrix}=I_{3}\]**

__Transpose of a matrix__

**If A is a matrix then transpose of the matrix A is obtained by interchanging the rows and columns . It is denoted by A'**

**Example**

\[If\; A=\begin{bmatrix} 17 & 8\\ 21& 23 \end{bmatrix}\; \; then\; \; A' =\begin{bmatrix} 17 &21 \\ 8& 23 \end{bmatrix}\]

\[If\; \; A=\begin{bmatrix} a &b &c \\ x& y & z\\ p &q &r \end{bmatrix}\; \; then,\; A'=\begin{bmatrix} a & x &p \\ b& y &q \\ c& z &r \end{bmatrix}\]

__Properties of the transpose of the matrices__

**i) (A')' = A, or Transpose of the transpose matrix is the matrix itself**

**ii) (kA)' = kA', where k is any scalar**

**iii) (A + B)' = A' + B'**

**Orthogonal Matrices**

Let A be a matrix and A' is its transpose.

A square matrix of order n is said to be orthogonal if its transpose is equal to its inverse.

or

A square matrix of order n is said to be orthogonal if product of its transpose and its inverse gives a unit matrix

Mathematically: A'A-1 = I

**Symmetric and Skew symmetric matrices**

**Symmetric Matrix :**

**A matrix A is said to be symmetric if A' = A**

**Skew symmetric matrix :**

**A matrix is said to be skew symmetric if A' = -A**

**Theorem 1:**

**For a square matrix A with real entries A + A' is always symmetric matrix and A - A' is always skew symmetric matrix.**

**Proof:**

**Let A + A' = B, then**

**B' = (A + A')' = A' + (A')' = A' + A = A + A' = B**

**⇒ B = A + A' is a symmetric matrix**

**Let C = A - A' , then**

**C' = (A - A')' = A' - (A')' = A' - A = - (A - A') = - C**

**Theorem 2:**

**Any square matrix can be expressed as the sum of symmetric and skew symmetric matrix.**

**Proof:**

**If A is a matrix and A' is a transpose of matrix A, then A + A' is always a symmetric matrix and A - A' is always a skew symmetric matrix**

**\[\Rightarrow P=\frac{1}{2}(A+A')\; is\; a\; symmetric\; matrix\]**

**\[\Rightarrow Q=\frac{1}{2}(A-A')\; is\; a\; skew\; \; symmetric\; matrix\]**

**\[A=P+Q=\frac{1}{2}(A+A')+\frac{1}{2}(A-A')=A\]**

**⇒ Every square matrix can be written as the sum of symmetric and skew symmetric matrix**

**Elementary operations of a matrix**

**There are mainly six operations on the matrix. Three of which are due to row and other three of which are due to column.**

**Here we explain only row operations. Column operations are also same.**

**i) Interchanging any two rows is denoted by**

**R**

_{i}**↔**

**R**

_{j}

_{}**ii) Multiplication of all the members of any row by a scalar k is is denoted by**

**R**

_{i}→ kR_{i}

_{}**iii) The addition of the elements of any row with the corresponding elements of any other row multiplied by any non-zero number is denoted by**

**R**

_{i}**→**

**R**

_{i}+ kR_{j}

**To find the inverse of the matrix by the method of elementary operations**

**To find A**

^{-1}by elementary operations we can either follow row operations or column operations.

**If we follow row operations then we write the equation A = IA**

**If
we follow column operations then we write the equation A = AI**

**Method of Finding the Inverse of 2 x 2 matrix
by Elementary Operations**

**Step 1: For Row operations we write A = IA**

**Step 2: Make a _{11} = 1**

**Step 3: With the help of a _{11} = 1 make a_{21} =
0**

**Step 4: Now make a _{22} = 1**

**Step 5: With the help of a _{22} = 1 make a_{12} =
0**

**Method of Finding the Inverse of 3 x 3 matrix
by Elementary Operations**

**Step 1: For Row operations we write A = IA**

**Step 2: Make a _{11} = 1**

**Step 3: With the help of a _{11} = 1 make a_{21 }and
a_{31} = 0**

**Step 4: Now make a _{22} = 1**

**Step 5: With the help of a _{22} = 1 make a_{12} and
a_{32} = 0**

**Step 6: Now make a _{33} = 1**

**Step 7: With the help of a _{33} = 1 make a_{13} and
a_{23} = 0**

__Invertible
matrices__

**If A and B are two square matrices of the same order such that AB =BA = I, then B is said to be inverse of A and is denoted by A**

^{-1}such that A^{-1 }= B. In this case A is said to be invertible.

**In other words :**

**If A is invertible then A**

^{-1}exists or If A^{-1}exists then A is said to be invertible.

**Theorem 3 :**

**Inverse of a matrix if exist then it is unique.**

**Proof:**

**If possible let us suppose that the matrix A has two inverse B and C**

**B is the inverse of A then AB = BA = I………… (i)**

**C is the inverse of A then AC = CA = I ………(ii)**

**From (i) and (ii) we get**

**AB = AC ⇒**

**B = C ………….(Cancelling A on both side)**

**⇒**

**Both the inverse are same.**

**Theorem 4:**

**If A and B are two invertible matrices of the same order then (AB)**

^{-1}= B^{-1}A^{-1}

**Proof :**

**By the definition of inverse we have**

**(AB)(AB)**

^{-1 }= I**Multiplying on both side by A**

^{-1 }we get

**A**

^{-1}(AB) (AB)^{-1}= A^{-1 }I**⇒**

**(A**

^{-1}A) B(AB)^{-1}= A^{-1}**(IB)(AB)**

^{-1}= A^{-1}**⇒**

**B(AB)**

^{-1}= A^{-1}**(B**

^{-1}B) (AB)^{-1 }= B^{-1}A^{-1}**⇒**

**I(AB)**

^{-1 }= B^{-1}A^{-1}

**⇒**

**(AB)**

^{-1 }= B^{-1}A^{-1}

^{}

__THANKS FOR WATCHING__

__PLEASE COMMENT BELOW__
🙏

- Get link
- Other Apps

### Breaking News

### Popular Post on this Blog

### Lesson Plan Maths Class 10 | For Mathematics Teacher

E-LESSON PLANNING FOR MATHEMATICS TEACHER CLASS 10TH lesson plan for maths class X cbse, lesson plans for mathematics teachers, Method to write lesson plan for maths class 10, lesson plan for maths class X, lesson plan for mathematics grade X, lesson plan for maths teacher in B.Ed. RESOURCE CENTRE MATHEMATICS LESSON PLAN (Mathematics) : CLASS 10 th Techniques of Making E-Lesson Plan : Click Here Click Here For Essential Components of Making Lesson Plan Chapter 1 : Number System This lesson plan is for the teachers who are teaching mathematics class 10 th For Complete Explanation Click Here New Lesson Plan with Technology Integration as suggested by CBSE in March, 2021 Class 10 Chapter 1 : Number System For Complete Explanation Click Here Chapter 2 : POLYNOMIALS This lesson plan is for the teachers who are teaching mathematics class 10 th For Complete Explanation Click Here Chapter 3 PAIR OF

### Lesson Plan Math Class 10 (Ch-1) | Real Numbers

E- LESSON PLAN SUBJECT MATHEMATICS CLASS 10 lesson plan for maths class 10 cbse, lesson plans for mathematics teachers, Method to write lesson plan for maths class 10, lesson plan for maths class 10 real numbers, lesson plan for mathematics grade 10, lesson plan for maths in B.Ed. TEACHER : SCHOOL : SUBJECT : MATHEMATICS CLASS : X STANDARD BOARD : CBSE LESSON TOPIC / TITLE : CHAPTER 1: REAL NUMBERS ESTIMATED DURATION: This lesson is divided into seven modules and it is completed in seven class meetings. PRE- REQUISITE KNOWLEDGE:- Number system of class IX Method of finding H.C.F. and L.C.M. of class VI and VII Ability to factorize the numbers ( Divisibility Test ) TEACHING AIDS:- Green Board, Chalk, Duster, Charts, Models etc. TECNOLOGY REQUIRED Smart Board, Projector, Laptop, Internet METHODOLOGY:- De

### Lesson Plan Maths Class XII | For Maths Teacher

Chapter-wise Maths Lesson Plan Class 12 Chapter wise Lesson Plan, Class XII Subject Mathematics for Mathematics Teacher. Effective way of Teaching Mathematics. Top planning by the teacher for effective teaching in the class. E lesson planning for mathematics. Click Here For Essential Components of Making Lesson Plan RESOURCE CENTRE MATHEMATICS LESSON PLAN : CLASS 12 BY CBSE MATHEMATICS Chapter 1 Relation & Functions This lesson plan is for the teachers who are teaching mathematics class 10+2 class . LESSON PLAN : CLASS 12 BY CBSE MATHEMATICS Chapter 2 Inverse Trigonometric Functions This lesson plan is for the teachers who are teaching mathematics class 10+2 class . LESSON PLAN : CLASS 12 BY CBSE MATHEMATICS Chapter 3 Metrices This lesson plan is for the teachers who are teaching mathematics class 10+2 class . LESSON PLAN : CLASS 12

## Comments

## Post a Comment